noetherian semigroup algebras and prime maximal orders

Noetherian semigroup algebras and

prime maximal orders

Isabel Goffa

AbstractLet S be a semigroup and K be a field. A K-space K[S], with basis Sand with multiplication extending, in a natural way, the operation on S,is called a semigroup algebra. It remains an open problem to characterizesemigroup algebras that are a prime Noetherian maximal order.

In this thesis, we give an answer to the problem for a large class of can-cellative semigroups and we illustrate these results with several examplesof concrete classes of Noetherian maximal orders. Indeed, we find neces-sary and sufficient conditions for a prime Noetherian algebra K[S] of asubmonoid S of a polycyclic-by-finite group to be a maximal order. Un-der an invariance condition on the minimal primes, our result is entirelyin terms of the monoid S and, in order to prove it, we describe the heightone prime ideals of K[S]. Recall that it is conjectured that polycyclic-by-finite groups G are the only groups having a Noetherian group algebra andK.A. Brown characterized when these group algebras K[G] are a primeNoetherian maximal order.

In case K[S] also satisfies a polynomial identity, this means in caseS is a submonoid of a finitely generated abelian-by-finite group, we showthat the invariance condition on the minimal primes of S is necessary forK[S] to be a prime Noetherian maximal order. Furthermore, we establisha general method for constructing non-abelian maximal order semigroupalgebras of finitely generated submonoids of abelian-by-finite groups, star-ting from abelian maximal orders. To obtain concrete constructions, wethus also need to deal with abelian finitely presented monoids A. If A hasa presentation with one or two defining relations, we determine necessaryand sufficient conditions for K[A] to be a domain that is a maximal order.The description is only in terms of the defining relations. Furthermore, wecompute the class groups of such semigroup algebras.

In the appendix, we briefly explain applications of maximal orders inspace-time coding. These applications softly point out that maximal ordersnot only might be interesting for experts in algebra, but also for specialistsin coding theory.

i

DankwoordMijn grootste dank ben ik verschuldigd aan mijn promotor Prof. Dr. EricJespers. Ontzettend dankbaar ben ik hem omdat hij mij de kans gaf ditdoctoraat te schrijven. Het is bewonderenswaardig hoe ver en diep zijnkennis over dit onderzoek reikt en hoe geduldig, goed en vriendelijk hijdie duidelijke kijk met mij wist te delen. De vele congressen en studiever-blijven die ik met hem heb mogen meemaken waren telkens zeer leerrijk enbijzonder aangenaam. Dankzij hem kreeg ik de kans talrijke interessante ensympathieke wiskundigen te ontmoeten en de mooie herinnering aan menigamusante congresavond zal mij altijd bijblijven.

I would like to warmly thank Prof. Dr. Jan Okninski. Thank youfor the cooperation and for the willingness to extensively answer to all myquestions. I am grateful for the papers we wrote together with Eric and forthe interesting conversations we had.

Bedankt aan het Instituut voor de Aanmoediging van Innovatie doorWetenschap en Technologie in Vlaanderen, dat mij het mogelijk maakteom vier jaar lang onderzoek te kunnen verrichten aan de Vrije UniversiteitBrussel. Telkens was alles tot in de puntjes geregeld en ik ben blij dat ikvia hen de uiterst vriendelijke Prof. Dr. Alfons Ooms heb leren kennen.

Een thesis schrijf je niet alleen. Op mijn weg ben ik tal van mensentegengekomen, die door hun blijk van interesse, er telkens opnieuw in slaag-den mij aan te sporen. Zo ben ik de collega’s van het wiskunde departementeen speciaal woordje van dank verschuldigd. Een koffiepauze zo nu en dan,een gezellige lunch, het zijn de dingen die elke werkdag telkens opnieuwaangenaam maken. De klasgenootjes tijdens mijn licentiaatsopleiding, dielater mijn collega’s werden, zou ik willen danken voor alle ontspannendemomenten en voor de vele motiverende babbels die we samen hadden. Dankook aan Philippe, die interessante seminaries gaf over space-time coding.

Mijn familie en vrienden wil ik uitermate danken omdat ze er steedsvoor me zijn en omdat ik altijd op hen kan rekenen. Joris, omdat je bentwie je bent en omdat je mij steunt in alles wat ik doe, ben ik jou ontzettenddankbaar. Isabel Goffa

Februari 2008

iii

Do not spoil what you have by desiring what you have not; but rememberthat what you now have was once among the things you only hoped for.

Epicurus

Contents

Abstract i

Acknowledgements in Dutch iii

1 Introduction 1

2 Prerequisites on semigroup and ring theory 72.1 Semigroups and semigroup algebras . . . . . . . . . . . . . . 72.2 Prime ideals in Noetherian and graded rings . . . . . . . . . 122.3 Polycyclic-by-finite groups and their algebras . . . . . . . . 172.4 Localization and Goldie’s theorem . . . . . . . . . . . . . . 192.5 Maximal orders, Krull orders and class groups . . . . . . . . 22

3 Maximal order semigroup algebras 293.1 Introduction and motivation . . . . . . . . . . . . . . . . . . 293.2 Results on height one prime ideals of K[S] . . . . . . . . . . 343.3 Applications of primes: Schelter’s theorem for semigroups . 403.4 Algebras of submonoids of polycyclic-by-finite groups . . . . 443.5 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4 Maximal order semigroup algebras satisfying a polynomialidentity 514.1 Algebras of submonoids of abelian-by-finite groups . . . . . 524.2 Constructing examples of maximal orders . . . . . . . . . . 544.3 An illustration of the main theorem . . . . . . . . . . . . . 55

5 Abelian maximal orders and class groups 675.1 Finitely presented maximal orders with one relation . . . . 685.2 Finitely presented maximal orders with two relations . . . . 755.3 Comments and problems . . . . . . . . . . . . . . . . . . . . 98

vii

viii

6 Examples of maximal orders 1056.1 Monoids of I-type . . . . . . . . . . . . . . . . . . . . . . . . 1066.2 Monoids of IG-type . . . . . . . . . . . . . . . . . . . . . . . 1086.3 Other examples . . . . . . . . . . . . . . . . . . . . . . . . . 123

A Maximal order applications in Space-Time Coding 127A.1 Space-Time Coding . . . . . . . . . . . . . . . . . . . . . . . 128A.2 Comments and further problems . . . . . . . . . . . . . . . 130

Summary in Dutch 133

Bibliography 139

Index 145

Contents

Chapter 1

Introduction

The aim of the work in this thesis is to present new classes of algebras with arich algebraic structure, such as prime Noetherian maximal orders. Becauseof the role of Noetherian rings in the algebraic approach in noncommutativegeometry, new concrete classes of finitely presented algebras recently gaineda lot of interest. Via the applications to the solutions of the Yang-Baxterequation, one such class also widens the interest into other fields, such asmathematical physics, and therefore these algebras have been extensivelystudied, see for example [20, 21, 25, 27, 39, 68].

In this thesis, constructions are given of finitely presented algebras overa field K defined by monomial relations. In order to describe when thesealgebras are Noetherian maximal orders, it is shown that the underlyinghomogeneous information determines the algebraic structure of the algebra.Hence, it is natural to consider such algebras as semigroup algebras K[S]and to investigate the structure of the monoid S. Furthermore, it turns outthat many of these algebras are subalgebras of group algebras of polycyclic-by-finite groups. This link to group algebras, that is of interest, yields a richresource of constructions not only for the non-commutative ring theorists,but also to researchers in semigroup theory and certain aspects of grouptheory.

In general, it remain unsolved problems to characterize when an arbi-trary semigroup algebra K[S] is Noetherian and when it is a prime Noethe-rian maximal order. Recall that the former question even for group algebrashas been unresolved. The only class of groups for which a positive answerhas been given is that of the polycyclic-by-finite groups, i.e. it is conjec-tured that these groups are the only groups that yield a Noetherian groupalgebra.

1

2

Brown ([6, 7]) characterized when such group algebras are prime Noethe-rian maximal orders and when they are unique factorization rings or do-mains, in the sense as studied by Chatters and Jordan ([12]).

Hence, in the context of semigroup algebras, it is natural to considersubmonoids of polycyclic-by-finite groups. Jespers and Okninski [45] charac-terized when the semigroup algebra of a submonoid of a polycyclic-by-finitegroup is right Noetherian (Theorem 3.1.4). It turns out that these semi-group algebras are right Noetherian if and only if they are left Noetherianand that, if K[S] is right Noetherian, the semigroup S is finitely generated.For general semigroup algebras it is not known whether K[S] right Noethe-rian implies S finitely generated. In [60], the author gives some classes ofsemigroups that satisfy this implication. Furthermore, if S is an abelianmonoid, Gilmer [28] proved that the semigroup algebra K[S] is Noetherianif and only if the semigroup S is finitely generated.

Concerning the maximal order problem, if S is an abelian monoid, An-derson [1, 2] proved that K[S] is a prime Noetherian maximal order ifand only if S is a submonoid of a finitely generated torsion-free abeliangroup and S is a maximal order in its group of quotients. More generally,Chouinard [14] proved that a commutative monoid algebra K[S] is a Krulldomain if and only if S is a submonoid of a torsion-free abelian group whichsatisfies the ascending chain condition on cyclic subgroups and S is a Krullorder in its group of quotients. In particular, it turns out that the heightone prime ideals of K[S] determined by the minimal primes of S are crucialand that the class group of K[S] is isomorphic to the class group of S.

Apart from the cases mentioned above, an answer to the question hasbeen obtained only for some special classes of cancellative semigroups, suchas submonoids of torsion-free finitely generated abelian-by-finite groups([41]), or for some special classes of maximal orders, such as principal idealrings ([38, 49]) and unique factorization rings ([47, 48]).

In this thesis we continue these investigations, we investigate when asemigroup algebra of a submonoid of a polycyclic-by-finite group is a primeNoetherian maximal order and we describe height one prime ideals of suchsemigroup algebras. So, for a large class of cancellative semigroups, we givean answer to the open question posed above.

We now briefly outline the content of each chapter. Chapter 2 is in-cluded for the convenience of the reader. This chapter contains the mainprerequisites of semigroup and ring theory used in the thesis.

Introduction

3

In Chapter 3 we elaborate on our joint work with Jespers and Okninskiand we characterize, under an invariance condition on the minimal primes,Noetherian semigroup algebras that are prime maximal orders, for sub-monoids of polycyclic-by-finite groups (Theorem 3.4.3). Knowledge of primeideals turns out to be crucial and therefore we first describe the height oneprime ideals of such prime Noetherian semigroup algebras (Theorem 3.2.2).If K[S] is a domain, i.e. if SS−1 is torsion-free, the same information onheight one primes, and even information on primes that are not of heightone, was obtained earlier by Jespers and Okninski in [42, 45]. However, weshow in this thesis that the information on primes that are not of heightone does not remain valid in case K[S] is prime and not a domain (Exam-ple 4.3.2).

As a consequence of the description of the height one prime ideals, weestablish going up and going down properties between prime ideals of S andprime ideals of S ∩H, where H is a subgroup of finite index in G = SS−1

(Corollary 3.2.5). These are the analogs of the important results known onthe prime ideal behaviour between a ring graded by a finite group and itshomogeneous component of degree e (the identity of the grading group). Asan application, it is shown that the classical Krull dimension clKdim(K[S])of K[S] is the sum of the prime dimension of S and the plinth length of theunit group U(S) (Corollary 3.3.1). Also, a result of Schelter is extendedto the monoid S: the prime dimension of S is the sum of the height anddepth of any prime ideal of S (Proposition 3.3.3).

Furthermore, as a consequence of the characterization obtained in Chap-ter 3, we describe in Chapter 4 when a semigroup algebra of a submonoidof a polycyclic-by-finite group is a prime Noetherian maximal order thatsatisfies a polynomial identity, this means we characterize when a semigroupalgebra of a submonoid of a finitely generated abelian-by-finite group is aprime Noetherian maximal order (Theorem 4.1.3). This characterizationextends the earlier result, Theorem 3.1.6, in case the group of quotients istorsion-free. Furthermore, if the polycyclic-by-finite group in Theorem 3.4.3is abelian-by-finite, we show that the invariance condition on the minimalprimes in Theorem 3.4.3 is not only sufficient, but is also necessary. In Sec-tion 4.2, we illustrate Theorem 4.1.3 with an example of a prime Noetherianmaximal order semigroup algebra (Example 4.3.2). As said above, this ex-ample also shows that Theorem 3.2.2 cannot be extended to prime idealsof height exceeding one. The obtained results in Chapters 3 and 4 can befound in the papers [30] and [31].

Introduction

4

Theorem 4.1.3 reduces the problem of determining whenK[S] is a primeNoetherian maximal order to the algebraic structure of S. It hence providesa strong tool for constructing new classes of such algebras. For some exam-ples the required conditions on S can easily be verified, but on the otherhand, for some other examples this still requires substantial work. Thatis why we establish a useful general method for constructing nonabeliansubmonoids of abelian-by-finite groups that are maximal orders, startingfrom abelian maximal orders (Proposition 4.2.1).

Anderson, Gilmer and Chouinard characterized abelian finitely gene-rated monoids that are maximal orders ([1, 2, 14, 28]). Despite this nice anduseful structural characterization, the following remains a challenging prob-lem for a finitely generated submonoid S of a torsion-free abelian group:determine necessary and sufficient conditions on the defining relations for Sto be a maximal order. In Chapter 5, we precisely solve this problem in caseS is defined by one or two relations (Theorem 5.1.2 and Theorem 5.2.2).So, in contrast to the structural description obtained by Anderson, Gilmerand Chouinard, our contribution is a computational approach (based onpresentations) to obtain a description of maximal orders defined via mono-mial relations. The advantage is that it also allows us to compute theclass group cl(S) of such monoids S and thus of their algebras K[S] (Theo-rem 5.1.4 and Theorem 5.2.4). This group is the basic tool in the study ofarithmetics of maximal orders ([23]) and hence it follows that our obtainedresults on abelian maximal orders and class groups, which are written inthe paper [32], might be interesting for experts in related areas.

In Chapter 6 we illustrate the theory developed in the preceding chap-ters on several concrete examples of semigroup algebras and we come backon the non-commutative algebras, mentioned in the beginning of this in-troduction, that share many properties with polynomial algebras in finitelymany commuting variables. Namely, we consider monoids of I-type, in-troduced by Gateva-Ivanova and Van den Bergh in [27], and later alsostudied by Jespers and Okninski in [43]. We reprove, with the help ofTheorem 4.1.3, that their semigroup algebras are Noetherian maximal or-ders and domains.

Furthermore, we generalize the concept of a monoid of I-type to thatof a monoid of IG-type and we characterize, again with the help of Theo-rem 4.1.3, when such semigroup algebras are prime Noetherian maximalorders. All the results on monoids of IG-type are published in our jointpaper [29] with Jespers. A few other examples of maximal orders are de-scribed in Chapter 6.

Introduction

5

Finally, in the Appendix, we explain applications of maximal orders inspace-time coding. We briefly outline the quite recently developed theoryof space-time codes and we explain how maximal orders can be used toobtain new results in the field. These applications softly point out thatmaximal orders not only might be interesting for experts in algebra, butalso for specialists in coding theory.

Introduction

6

Introduction

Chapter 2

Prerequisites on semigroupand ring theory

This chapter is devoted to the necessary background on semigroup, groupand ring theory. The reader who is familiar with all the required conceptscan omit this chapter. The material is a mixture of classical results thatcan be found in [45, 58, 60, 64, 65].

2.1 Semigroups and semigroup algebras

This section is an attempt to bring together the basic definitions and majorresults we will need from the theory of semigroups and semigroup algebras.Some notions in semigroup theory can be derived from group theory, butmost of them resemble ring theory more closely. Some notions are based onfamiliar concepts in group and ring theory. Our notation and terminologyfollows closely that of [15], [45] and [60].

A semigroup S is a non-empty set with an associative binary operation.We will denote this operation multiplicatively. If S has an identity 1 thenit is called a monoid and we write U(S) for the group {s ∈ S | sr = rs =1 for some r ∈ S}, which is called the unit group of S. A monoid withS = U(S) is simply a group. An element θ of a semigroup S is a zero of S ifit is a left and right zero, that is θs = sθ = θ for all s ∈ S. A null semigroupis a semigroup with zero in which the product of any two elements is thezero element θ.

If S is a semigroup without an identity, one can adjoin an identity bysimply adding a new element 1 and extending the multiplication by defining

7

2.1. Semigroups and semigroup algebras 8

1s = s1 = s for all s ∈ S ∪ {1}. We denote this new semigroup by S1. If Salready has an identity, then one agrees that S1 = S.

In a very similar way, one can always adjoin a zero element θ, and wewrite

S0 = S ∪ {θ}

for this new semigroup.A subsemigroup of a semigroup S is a non-empty subset which is closed

under multiplication. A submonoid is a subsemigroup which has an identity.A subgroup G of S is a subsemigroup which is a group. Note that even ifS is a monoid, then G does not necessarily have the same identity elementas S.

For a semigroup S, the subset

Z(S) = {x | xs = sx for all s ∈ S}

is called the centre of S and it is a commutative subsemigroup of S, thatis, any two elements commute.

We now recall the definition of a semigroup ring.

Definition 2.1.1. Let R be a ring and let S be a semigroup. The semigroupring R[S] is the ring whose elements are all formal sums

∑s∈S rss, with

each coefficient rs ∈ R and all but finitely many of the coefficients equalzero. Addition is defined component-wise so that∑

s∈S

rss+∑s∈S

qss =∑s∈S

(rs + qs)s.

Multiplication is given by(∑s∈S

rss

)(∑s∈S

qss

)=∑s∈S

∑uv=s; u,v∈S

ruqv

s.

If R = K is a field, then K[S] is called a semigroup algebra.

For an element α =∑

s∈S rss ∈ R[S], with each rs ∈ R, we putsupp(α) = {s ∈ S | rs 6= 0}, the support of α.

If S is a semigroup without a zero θ it is clear that

R[S0] = R[T ]×Rθ ∼= R[S]×Rθ,

Prerequisites on semigroup and ring theory


a direct product of rings, where T = {s − θ | s ∈ S} is a semigroupisomorphic with S. In the case S has a zero element θ, we write R0[S] forthe quotient ring

R[S]/Rθ.

The ring R0[S] is called the contracted semigroup ring of S over R. IfR = K is a field then K0[S] is called the contracted semigroup algebra.

Note that K0[S ∪ {θ}] ∼= K[S]. Hence every semigroup algebra can beseen as a contracted semigroup algebra.

Definition 2.1.2. If T is a non-empty subset of a monoid S, we write〈T 〉 for the submonoid generated by T and gr(T ) for the subgroup generatedby T . If T is finite, say T = {t1, . . . , tn}, we often write 〈t1, . . . , tn〉 orgr(t1, . . . , tn). A monoid S is cyclic if S = 〈x〉 for some x ∈ S. An elementx of a monoid S is a periodic element if 〈x〉 is finite.

Obviously, a semigroup homomorphism from a semigroup S to a semi-group T is a function f : S −→ T such that f(xy) = f(x)f(y) for allx, y ∈ S.

Let I be a non-empty subset of a semigroup S. Then I is a right idealof S if xs ∈ I for all s ∈ S and x ∈ I. A left ideal is defined similarly andI is an ideal of S if it is a left and right ideal of S. If a ∈ S, then the idealgenerated by a, that is, the smallest ideal of S containing a, is

Ja = S1aS1 = SaS ∪ Sa ∪ aS ∪ {a}.

We say that a semigroup S is simple if it has no ideals other than Sitself. This definition is not interesting in case S has a zero element θ,since {θ} always is an ideal of S. Therefore we say that a semigroup witha zero θ is 0-simple if S2 = S and S has no ideals other than S and {θ}.Recall that a semigroup with zero θ is a nil semigroup if every s ∈ S isnilpotent, so there exists k ∈ N0 with sk = θ. If Sk = {θ} for some k ∈ N0,then S is called nilpotent. A (left) ideal L of a semigroup with zero θ isnil if all its elements are nilpotent. L is said to be nilpotent if Lk = θ forsome k ∈ N0 (that is, l1 . . . lk = θ for all li ∈ L). The following theorem istaken from [22, Theorem 17.22]. It provides the analog for semigroups of aLevitzki-theorem.

Theorem 2.1.3. Let S be a monoid with a zero element. If S satisfies theascending chain condition on one-sided ideals then every nil ideal of S isnilpotent.



For a relation ρ on a set S and s, t ∈ S we write (s, t) ∈ ρ also as s ρ t.An equivalence relation ρ on a semigroup S is called a congruence if forany s, t, x in S we have sx ρ tx and xs ρ xt whenever s ρ t. We write S/ρfor the set of ρ-classes in S. It also is a semigroup for the natural operationinduced by that of S.

Definition 2.1.4. Let S be a semigroup, ρ a congruence relation and S/ρthe semigroup of ρ-classes in S. Denote by φρ : S −→ S/ρ the naturalsemigroup homomorphism. Its K-linear extension to an algebra epimor-phism K[S] −→ K[S/ρ] we also denote by φρ and its kernel is the idealI(ρ) of K[S] generated by the set {s− t | s, t ∈ S with s ρ t}. Hence

K[S/ρ] ∼= K[S]/I(ρ)

as K-algebras.

We give three examples of congruence relations that are relevant in ourwork. First, let S be a submonoid of a group G. Assume F is a normalsubgroup of G so that F ⊆ S. Denote by ∼F the relation on S defined asfollows:

s ∼F t if s = ft for some f ∈ F.

Because F is normal in G it is easily seen that ∼F is a congruence relationon S.

Second, let I be an ideal in an arbitrary semigroup S and let ρ denotethe congruence on S defined by s ρ t if s = t or s, t ∈ I. The semigroupS/ρ is usually denoted by S/I and is called the Rees factor of S by I. Also,for a field K, it is easily verified (see [60, Lemma 4.7]) that

K[S]/K[I] ∼= K0[S/I].

Intuitively, when we pass from S to S/I, we have identified all the elementsof I with zero. As a special case, it is convenient to agree that S/∅ = S,even though ∅ is not an ideal. There is a one-to-one correspondence betweenthe ideals of S containing I and the ideals of S/I.

Lemma 2.1.5. If I ⊆ J are ideals of S, then J/I is an ideal of S/I and(S/I)/(J/I) ∼= S/J .

If T is a subsemigroup of S and I is an ideal of S which intersects Tnontrivially, then T ∩ I is an ideal of T and T/(T ∩ I) ∼= (T ∪ I)/I.

Thirdly, we associate with an ideal J of K[S] a congruence relation ρJ

on S. So this is the opposite of the process explained in Definition 2.1.4.



Definition 2.1.6. Let S be a semigroup and let J be an ideal of K[S].Then, ρJ denotes the congruence on S defined by

ρJ = {(s, t) | s, t ∈ S, s− t ∈ J}.

Note that I(ρJ) ⊆ J . There are natural epimorphisms

K[S] −→ K[S/ρJ ]

andK[S/ρJ ] −→ K[S]/J.

One often identifies the semigroup S/ρJ with its image in K[S]/J .

Throughout this thesis we are mainly interested in submonoids S ofgroups. Often the elements of these groups are fractions of elements of S.In order to clearly describe this we need more terminology.

Definition 2.1.7. A semigroup S is called left cancellative (respectivelyright cancellative), if, for every a, b, x in S, xa = xb implies a = b (respec-tively ax = bx implies a = b). A semigroup is cancellative if it is left andright cancellative.

We say that S satisfies the right Ore condition if for every a, b ∈ S wehave aS ∩ bS 6= ∅. It is well known that, for cancellative semigroups S,this happens if and only if S has a group of right quotients G = SS−1 ={ab−1 | a, b ∈ S}, which is unique up to isomorphism. If S also has a groupof left quotients H (defined symmetrically), then G,H are isomorphic andare referred to as the group of (two-sided) quotients of S.

In the next lemma we determine a sufficient condition for a cancellativesemigroup to have a group of (two-sided) quotients ([45, Lemma 2.1.3]).

Lemma 2.1.8. Assume that S is a left cancellative semigroup with nofree nonabelian subsemigroups. Then, for every right ideals I, J of S wehave I ∩ J 6= ∅. In particular, if S is cancellative with no free nonabeliansubsemigroups, then S has a group of (two-sided) quotients.

From Lemma 2.1.8 it follows that a submonoid S of an abelian-by-finitegroup G has a group of quotients. Recall that a group is abelian-by-finiteif it has a normal abelian subgroup of finite index and thus, indeed, such amonoid can not have free nonabelian subsemigroups. In general, if P andQ are two group properties, a group G is said to be P -by-Q, if there existsa normal subgroup N of G that has property P and so that the quotientgroup G/N has property Q.


2.2. Prime ideals in Noetherian and graded rings 12

Finally, we define the dimension of a semigroup, but before we do this,we give some more definitions. A semigroup S (with a zero element) isprime if IJ = {θ} for some ideals I, J of S implies that I = {θ} orJ = {θ}. A proper ideal Q of a semigroup S (with a zero element) is aprime ideal if S/Q is a prime semigroup. Note that, by definition, primeideals are different from S. A minimal prime ideal in S is a prime ideal ofS that does not properly contain another prime ideal.

The prime spectrum Spec(S) is the set of all prime ideals of S. The rankrk(S) of a monoid S (not necessarily cancellative) is the supremum of theranks of the free abelian subsemigroups of S (see for example [60]). So, fora finitely generated torsion-free abelian group A this corresponds with theclassical notion of torsion-free rank of A. If S does not have a zero elementthen the height ht(Q) of a prime ideal Q is the maximal length n of a chainQ0 ⊂ Q1 ⊂ Q2 ⊂ · · · ⊂ Qn = Q, where Q0 = ∅ and Q1, . . . , Qn−1 are primeideals of S. So, in this case, ht(Q) = 1 if and only if Q is a minimal primeideal of S. On the other hand, if S has a zero element, then ht(Q) is themaximal length of such a chain with all Qi prime ideals of S, i ≥ 0. If suchn does not exist then we say that the height of Q is infinite. By X1(S) wedenote the set of all minimal prime ideals of S.

Definition 2.1.9. The dimension dim(S) of S is defined in the followingway. By definition dimS = 0 if S = {e}. If S has no zero element, thendim(S) is the maximal length n of a chain Q0 ⊂ Q1 ⊂ Q2 ⊂ · · · ⊂ Qn,where Q0 = ∅ and Qi are prime ideals of S for i > 0, or ∞ if such n doesnot exist. If {e} 6= S has a zero element, then dim(S) is the maximal lengthn of such a chain with all Qi (i ≥ 0) prime ideals of S, or ∞ if such n doesnot exist.

2.2 Prime ideals in Noetherian and graded rings

In this section we recall some basic terminology and some well known funda-mental structural results on prime ideals in Noetherian and graded rings.Several ring dimensions, such as classical Krull dimension and Gelfand-Kirillov dimension will be defined. For a survey and more detail we referthe reader to [45], [58] and [65].

A ring R will mean a nonzero associative ring, usually assumed to havean identity 1, except specified otherwise. Ring homomorphisms will beexpected to preserve the identity and subrings to have the same identity.

A ring R is said to be right Noetherian (respectively right Artinian) ifit satisfies the ascending (respectively descending) chain condition on right



ideals. Similarly one defines left Noetherian and left Artinian rings. ANoetherian (respectively Artinian) ring is one that is both right and leftNoetherian (respectively Artinian). A right Artinian ring is right Noethe-rian.

Recall that a ring R is said to be prime if the product of two nonzeroideals is nonzero. Obviously, in this case, the centre Z(R) of R is a domain,i.e. it has no zero-divisors. A proper ideal P of R is prime if R/P is aprime ring. A minimal prime ideal in R is a prime ideal of R that does notproperly contain another prime ideal. Any prime ideal is contained in amaximal ideal and maximal ideals are prime ideals. A ring R is said to besimple if R has precisely two ideals, that is, {0} is a maximal ideal. A ringR is (right) semisimple if it is semisimple as a right R-module. A nonzeromodule M is said to be simple, if {0} and M are its only submodules. Amodule that is the direct sum of simple submodules is called semisimple.A ring R is said to be semi-local if R/J(R) is semisimple, where J(R) isthe Jacobson radical of R, i.e. the intersection of all maximal left ideals ofR. Every nil ideal of R is contained in J(R). A (left) ideal L of a ring Ris nil if all its elements are nilpotent. L is said to be nilpotent if Ln = 0for some n ∈ N0 (that is, l1 . . . ln = 0 for all li ∈ L). The prime radical (orBaer radical) B(R) of a ring R is the intersection of the prime ideals of R.A ring R is semiprime if and only if B(R) = {0}, and a proper ideal P ofR is semiprime if R/P is a semiprime ring.

Theorem 2.2.1. (Levitzki) If R is a left Noetherian ring, then every nilleft or right ideal is nilpotent.

The prime spectrum Spec(R) of a ring R is the set of all prime idealsof R. The height ht(P ) of a prime ideal P of R is the largest length of achain of prime ideals contained in P . So ht(P ) = 0 means P is a minimalprime ideal. By X1(R) we denote the set of all height one prime ideals ofR.

The following theorem, which is called the principal ideal theorem ([58,Theorem 4.1.11, Corollary 4.1.12]), concerns height of primes in right Noethe-rian rings. This theorem will often be used in our further work. Recall thatan element n of a ring R is a normal element if Rn = nR.

Theorem 2.2.2. Let R be a right Noetherian ring and n a normal elementof R which is not a unit. If P is a prime ideal of R minimal over Rnthen ht(P ) ≤ 1. More general, if Q is any prime ideal containing n thenht(Q) ≤ ht(Q/Rn) + 1.



The classical Krull dimension clKdim(R) of R is the supremum of thelengths of all finite chains of prime ideals or ∞ if such chain does not exist.We next recall the Gelfand-Kirillov dimension of an algebra. It measuresthe rate of growth of K-algebras in terms of any generating set and for asemigroup algebra K[S] it measures the rate of growth of the semigroup S.

Definition 2.2.3. Suppose R is as a K-algebra generated by a finite di-mensional subspace V . Put R0 = V 0 = K and Rn =

∑ni=0 V

i for eachpositive integer n. Clearly each Rn is a finite dimensional K-space. Thenumber

limsupn→∞

(log dimKRn

log n

),

is independent of the choice of V . It is called the Gelfand-Kirillov dimensionof R and it is denoted by GK(R). Now GK(R) = 0 means that R is a finitedimensional K-algebra. Furthermore, GK(R) is finite if and only if thereexists a positive integer m so that dimKRn ≤ nm for all sufficiently largepositive integers n. For a not necessarily finitely generated K-algebra Rone defines the Gelfand-Kirillov dimension as the supremum of all GK(R

′),

where R′runs through all finitely generated K-subalgebras of R.

If K is a field and R a finitely generated commutative K-algebra, thenGK(R) = clKdim(R).

In case a semigroup algebra K[S] of a cancellative semigroup S satisfiesa polynomial identity, there is not only a nice link between the classi-cal Krull dimension and the Gelfand-Kirillov dimension of K[S], but alsobetween these dimensions and the rank of S (see for example [60, Theo-rem 23.4]). Recall that an algebra R over a field K satisfies a polynomialidentity (abbreviated, R is a PI algebra) if there exists a nonzero polyno-mial f(x1, . . . , xn) in the free K-algebra on generators x1, . . . , xn so thatf(r1, . . . , rn) = 0 for all r1, . . . , rn ∈ R.

Theorem 2.2.4. Let S be a cancellative semigroup and K a field. If K[S]satisfies a polynomial identity then clKdim(K[S]) = rk(S) = GK(K[S]).

We now recall a characterization of semigroup algebras of cancellativesemigroups satisfying a polynomial identity (see [45, Theorem 3.1.9]).

Theorem 2.2.5. Let K be a field. If S is a cancellative semigroup thenK[S] satisfies a polynomial identity if and only if S has a group of quotientsG so that K[G] satisfies a polynomial identity. The latter holds if and only ifG is abelian-by-finite and char(K) = 0 or G is (finite p-group)-by-(abelian-by- finite) and char(K) = p > 0.



Another important theorem concerning height of primes, is Schelter’stheorem (see for example [58, Theorem 13.10.12]). An affine K-algebrais an algebra that is finitely generated as a K-algebra by a finite set ofelements or, equivalently, by a finite dimensional K-subspace V .

Theorem 2.2.6. (Schelter) Let R be a prime PI affine K-algebra andP ∈ Spec(R). Then clKdim(R) = ht(P ) + clKdim(R/P ).

From Schelter’s theorem the following corollary easily follows. Recallthat a ring R has the catenary property if given any P, P

′ ∈ Spec(R) withP

′ ⊆ P , any two saturated chains of primes between P and P′

have thesame length. A chain is saturated if no additional term can be inserted.

Corollary 2.2.7. Each PI affine K-algebra has the catenary property.

To end this section, let us concentrate on prime ideals in graded rings.Since several of the proofs on semigroup algebras given in this thesis even-tually rely on reductions to some graded rings, we need to recall somestructural results in this general framework. There is a strong link betweenthe primes of a finite group graded ring and those of its identity component(see [65]). However, we only state here the properties relevant for later use.

Let G be a group. A ring R is said to be G-graded if R = ⊕g∈GRg,the direct sum of additive subgroup Rg of R such that RgRh ⊆ Rgh, forall g, h ∈ G. If always RgRh = Rgh then R is said to be strongly graded.The additive groups Rg are called the homogeneous components of R andRe (also often denoted as R1) is the identity component, where e is theidentity of the group G. The elements of ∪g∈GRg are called homogeneous.Obviously a group algebra K[G] is an example of a strongly G-graded ring.This ring, however, has many other natural gradings. For example if Nis a normal subgroup of G and T is a transversal of N in G then K[G] isstrongly G/N -graded with homogeneous components K[N ]t where t ∈ T .Actually, K[G] = K[N ] ∗ (G/N), a crossed product of G/N over K[N ] (see[65] for the definition).

The following theorem is taken from [65, Theorem 17.9].

Theorem 2.2.8. Let R = ⊕g∈GRg be a ring graded by a finite group Gwith identity e.

1. Cutting Down If P ∈ Spec(R) then

P ∩Re = Q1 ∩ · · · ∩Qn,

where n ≤ |G| and Q1, · · · , Qn are all the prime ideals of Re minimalover P ∩ Re. Furthermore, each prime Qi has the same height as Pand P is maximal among all ideals I of R such that I ∩Re ⊆ Qi.



2. Lying Over If Q ∈ Spec(Re) then there exists P ∈ Spec(R) (of thesame height as Q) so that

P ∩Re = Q1 ∩ · · · ∩Qn,

for some primes Q = Q1, · · · , Qn of Re minimal over P ∩ Re (againall of the same height as Q), and there are at most |G| such primeideals. One says that P lies over Q.

3. Incomparability Given the lying over diagram,

P1

↙...

Q1 P2... ↙Q2

where P1 lies over Q1, P2 lies over Q2 and P2 ⊆ P1, Q2 ⊆ Q1. IfP1 6= P2, then Q1 6= Q2.

4. Going up Suppose Q2 is a prime ideal of Re and P2 is a prime idealof R lying over Q2.

(a) If Q1 is a prime ideal of Re containing Q2 then there exists aprime ideal P1 of R lying over Q1 so that P2 ⊆ P1.

(b) If P1 is a prime ideal of R containing P2 then there exists aprime ideal Q1 of Re containing Q2 so that P1 lies over Q1.

5. Going down Suppose Q1 is a prime ideal of Re and P1 is a primeideal of R lying over Q1.

(a) If Q2 is a prime ideal of Re contained in Q1 then there exists aprime ideal P2 of R lying over Q2 so that P2 ⊆ P1.

(b) If P2 is a prime ideal of R contained in P1 then there exists aprime ideal Q2 of Re contained in Q1 so that P2 lies over Q2.

Furthermore clKdim(R) = clKdim(Re).


2.3. Polycyclic-by-finite groups and their algebras 17

2.3 Polycyclic-by-finite groups and their algebras

In this section we state some important facts on polycyclic-by-finite groupsand their group algebras. Basic references for such groups are [50, 64, 69].

The concept of a polycyclic-by-finite group yields a non-commutativegeneralization of a finitely generated abelian group.

Definition 2.3.1. A group G is said to be polycyclic-by-finite if it has asubnormal series

{1} = G0 ⊆ G1 ⊆ · · · ⊆ Gn−1 ⊆ Gn = G,

(so Gi / Gi+1, for 0 ≤ i ≤ n−1) such that G/Gn−1 is finite and every otherfactor Gi/Gi−1 is cyclic. Equivalently, a group G is polycyclic-by-finite, ifthere exists a normal subgroup N of finite index in G that is either trivialor poly-infinite-cyclic, i.e. N has a subnormal series with infinite cyclicfactors.

The number of infinite cyclic factors is independent of the choice of N .It is called the Hirsch rank of G and it is denoted by h(G).

If K is a field and G is an arbitrary group then the group algebraK[G] has an involution that K-linearly extends the classical involution onG which maps g to g−1. It follows that K[G] is right Noetherian if andonly if it is left Noetherian. Moreover, if K[G] is right Noetherian, thenG satisfies the ascending chain condition on subgroups, in particular, G isfinitely generated.

It remains an open problem to characterize when a group algebra isNoetherian. In case G is polycyclic-by-finite, an easy proof by induction onthe Hirsch rank yields the following result ([64, Corollary 10.2.8]); it givesthe only known class of groups for which the group algebra is Noetherian.

Theorem 2.3.2. Let K be a field. If G is a polycyclic-by-finite group thenthe group algebra K[G] is Noetherian.

For an arbitrary group G the set of torsion (we also say, periodic)elements is denoted by G+. If G+ = {1} then G is said to be torsion-free.It is well known that, if the group algebraK[G] is a domain, i.e. K[G] has nozero divisors, then G is torsion-free. Recall that the finite conjugacy centre(FC-centre for short) ∆(G) of a group G is the set of those elements g ∈ Gthat have only finitely many conjugates, or equivalently the centralizerCG(g) of g in G has finite index. The set (∆(G))+ will be denoted as∆+(G). Both sets are characteristic subgroups of G and ∆(G)/∆+(G) is


2.3. Polycyclic-by-finite groups and their algebras 18

a torsion-free abelian group ([64, Lemma 4.1.6]). Clearly, a finite normalsubgroup is contained in ∆+(G) and the latter group is trivial if and only ifno such nontrivial subgroups exist. A group G so that ∆(G) = G is calleda finite conjugacy group, or FC-group for short.

The following theorem states when the group algebra of a polycyclic-by-finite group is prime and when it is a domain (see [64, Theorem 4.2.10]and [65, Theorem 37.5]).

Theorem 2.3.3. Let K be a field and G a polycyclic-by-finite group. Thenthe group algebra K[G] is prime if and only if ∆+(G) = {1}. Furthermore,the group algebra K[G] is a domain if and only if the group G is torsion-free.

We also mention the following result, which will be used often in thisthesis and which can be found in ([60, Theorem 7.19]).

Theorem 2.3.4. Let K be a field. If S is a monoid with a group of rightquotients G = SS−1 then the semigroup algebra K[S] is prime if and onlyif K[G] is prime. Furthermore, K[S] is a domain if and only if K[G] is adomain.

Through the work of Roseblade (see for example [64, 65]), group alge-bras K[G] of polycyclic-by-finite groups G over a field K are rather wellunderstood. In order to state the required properties some more notationand terminology is needed.

Suppose a group H acts on a set A. If a ∈ A and h ∈ H then we denotethe action of h on a as ah. An element a ∈ A is said to be H-orbital (or,simply, orbital) if its H-orbit is finite. The set of H-orbital elements in Awe denote by ∆H(A). Thus

∆H(A) = {a ∈ A | [H : CH(a)] <∞}

withCH(a) = {h ∈ H | ah = a}.

Definition 2.3.5. Let H be a group. A finite dimensional Q[H]-moduleV is said to be a rational plinth for H if V is an irreducible Q[H1]-modulefor all subgroups H1 of finite index in H. Now let H act on a finitelygenerated abelian group A. Then A is a plinth for H if, in additive notation,V = A⊗Z Q is a rational plinth.

Thus A is a plinth if and only if no proper pure subgroup of A is H-orbital. Recall that a subgroup B of A is pure if nA ∩ B = nB for allintegers n. The plinth A is centric if and only if the centralizer CH(A) isof finite index in H, or equivalently, A has rank one.


2.4. Localization and Goldie’s theorem 19

A normal series

{1} = G0 ⊆ G1 ⊆ · · · ⊆ Gn = G

(every Gi / G) for the polycyclic-by-finite group G is called a plinth seriesfor G if each quotient Gi/Gi−1 is either finite or a plinth for G. It is notnecessarily true that every G has a plinth series. However, any polycyclic-by-finite group G has a normal subgroup N of finite index with a plinthseries, say

{1} = N0 ⊆ N1 ⊆ · · · ⊆ Nn = N.

The number of infinite factors Ni/Ni−1 is called the plinth length of G andis denoted by pl(G). This parameter is independent of the choice of N andof the particular series for N . If F is a normal subgroup of G so that G/Fis abelian then pl(G) ≤ pl(F ) + rk(G/F ) (recall that rk(B) denotes thetorsion free rank of an abelian group B).

We now can formulate a fundamental result on prime ideals of groupalgebras of polycyclic-by-finite groups ([65, Theorem 19.6]).

Theorem 2.3.6. Let G be a polycyclic-by-finite group and K a field. Thefollowing properties hold.

1. clKdim(K[G]) = pl(G).

2. If K is absolute (that is, K is algebraic over a finite field) then everyright primitive ideal M of K[G] is maximal and K[G]/M is finitedimensional.

We finish with one more result ([53]).

Theorem 2.3.7. A group algebra of a polycyclic-by-finite group is catenary.

2.4 Localization and Goldie’s theorem

We will now study Ore sets and localizations at prime ideals. We stateGoldie’s theorem and we mention the behaviour of prime ideals under somelocalized rings. For more information about localization at prime ideals innon-commutative rings, we refer the reader to [33, 58].

For a subsetX of a ringR (or more general for a subsetX of a semigroupwith zero) we put raR(X) = {r ∈ R | Xr = {0}}, the right annihilator ofX in R. If X = {x} then we simply denote this set by raR(x). One saysthat an element x is right regular if raR(x) = {0}. Similarly one defines the



left annihilator laR(X) and left regular elements. An element that is bothright and left regular is said to be regular. Clearly, a non-regular elementis a zero divisor and all elements in a domain are regular.

A multiplicatively closed subset X of a ring R is said to be a right Oreset if xR ∩ rX 6= ∅ for any x ∈ X and r ∈ R. If X consists of regularelements then RX−1 denotes the ring of right quotients of R with respectto X. This ring will be called the localization of R with respect to X. If,moreover, X is the set consisting of all regular elements of R then RX−1

is called the (classical) ring of right quotients of R and it is denoted byQcl(R), or simply Q(R). So

Qcl(R) = {rx−1 | r, x ∈ R, x regular}.

In case the set of regular elements is a right Ore set one says that R hasa classical ring of right quotients. The left versions of these notions aredefined similarly.

Example 2.4.1. Let R be a commutative Noetherian ring and P a primeideal of R. Then the set R\P is an Ore set and the localization R(R\P )−1

is a local ring (that will be denoted by RP ) with unique maximal ideal PRP .

For non-commutative rings, much of this technique is not available.Localization at a prime ideal can be impossible. Suppose R is a non-commutative ring. If P is a completely prime ideal, i.e. R/P is an integraldomain, then R \ P is a multiplicatively closed (m.c. for short) subset ofR. However for a general prime ideal P of R, R/P may have zero divisorsand then R \ P is not a m.c. set. There is a natural m.c. set associatedwith an ideal I:

CR(I) = {s ∈ R | [s+ I] is regular in R/I}.

Clearly, CR(0) is the set of all regular elements of R.

Definition 2.4.2. A prime ideal P of a Noetherian ring R is said to belocalizable if CR(P ) is an Ore set. In this case, one can form the par-tial quotient ring RP , by localizing at CR(P ). The localization RP is aNoetherian local ring with unique maximal ideal PRP .

Although forming a non-commutative ring of fractions is not alwayspossible, for Noetherian rings, Goldie’s theorem shows that the formationof a ring of fractions is always possible, not merely for an integral domain,



but for any semiprime ring. Moreover, the ring of fractions is a semisim-ple Artinian ring. But before we state this theorem, we need some moredefinitions.

A ring R is said to satisfy the ascending chain condition on direct sum-mands (of left ideals) if there do not exist strict ascending chains of theform

L1 ⊂ L1 ⊕ L2 ⊂ L1 ⊕ L2 ⊕ L3 ⊂ · · ·with every Li a left ideal of R. The (unique) number of direct summands iscalled the uniform dimension of R (or Goldie dimension). One says that aring R satisfies the ascending chain condition on left annihilators if everyascending chain left annihilators stabilizes (the latter are left ideals of theform laR(X) with X ⊆ R).

Definition 2.4.3. A ring R is (left) Goldie if R satisfies both the ascendingchain condition on left annihilators and the ascending chain condition ondirect summands of left ideals. Similarly one defines a right Goldie ring.A ring that is both right and left Goldie is simply called a Goldie ring.

Clearly, a Noetherian ring is an example of a Goldie ring.We are now ready for Goldie’ s theorem [58, Theorem 2.3.6]. Recall

that a left ideal L of a ring R is said to be left essential if L∩X 6= {0} forany nonzero left ideal X of R.

Proposition 2.4.4. (Goldie) The following conditions are equivalent fora ring R.

1. R is a semiprime Goldie ring.

2. R has a classical ring of quotients which is semisimple Artinian.

3. A left ideal is essential if and only if it contains a regular element.

In particular, the ring R is prime Goldie if and only if R has a classicalring of quotients which is simple Artinian.

In particular, it follows from Goldie’ s theorem that a prime Noetherianring has a simple Artinian classical ring of quotients. The following theo-rem, which follows immediately from the Wedderburn-Artin theorem (seefor example [58, Theorem 0.1.10]), describes such simple Artinian rings ofquotients.

Theorem 2.4.5. A ring R is a simple Artinian ring if and only if R ∼=Mn(D) for some (uniquely determined) natural number n and a divisionring D.


2.5. Maximal orders, Krull orders and class groups 22

To end this section, we consider the behaviour of prime ideals undersome ring extensions. Again we only state one property that is relevant forlater use (see [33, Theorem 9.20, Lemma 9.21, Theorem 9.22]).

Theorem 2.4.6. Let X be a right Ore set consisting of regular elements ina ring R. Assume that R is right Noetherian or that R satisfies a polynomialidentity and RX−1 is right Noetherian. Then the following properties hold.

1. If I is an ideal of R then Ie = IRX−1 is an ideal of RX−1.

2. The maps I → Ie and J → J ∩ R are inverse bijections between theset of prime ideals of RX−1 and the set of those prime ideals of Rthat intersect X trivially.

2.5 Maximal orders, Krull orders and classgroups

It is well known that integrally closed Noetherian domains, or more gener-ally, Krull domains ([23]), are of fundamental importance in several areas ofmathematics. In this section we concentrate on prime Noetherian maximalorders, Krull orders, Asano orders and unique factorization rings; these aregeneralizations to the class of noncommutative rings. In the vast literatureone can find several types of such generalizations. Some of the relevantreferences for our purposes are [9, 10, 45, 52, 55, 56, 57, 71]. For a surveywe refer the reader to [58].

Let R be a prime Goldie ring with classical ring of quotientsQ = Qcl(R).One simply says that R is an order in Q. So, by Goldie’s theorem 2.4.4,every regular element of R is invertible in Q and elements in Q can bewritten as r1c−1

1 = c−12 r2, for some r1, r2, c1, c2 ∈ R with c1, c2 regular. If

T is a subring of Q and a, b are invertible elements in Q so that aTb ⊆ Rthen also T is an order in Q. This leads to an equivalence relation on theorders in Q.

Definition 2.5.1. One says that two orders R1 and R2 in Q are equivalentif there are units a1, a2, b1, b2 ∈ Q so that a1R1b1 ⊆ R2 and a2R2b2 ⊆ R1.The order R is said to be maximal if it is maximal within its equivalenceclass.

A commutative integral domain D with field of fractions K is a maximalorder if D is completely integrally closed, that is, if x, c ∈ K with c 6= 0and cD[x] ⊆ D then x ∈ D. Clearly, a completely integrally closed domain



is integrally closed , that is, if x ∈ K is so that D[x] is a finitely generatedD-module then x ∈ D. If D is a Noetherian domain then both notions areequivalent.

The study of maximal orders is aided by the notion of a fractional ideal.So suppose R is an order in its classical ring of quotients Q. A fractionalR-ideal (or simply, fractional ideal if the order R is clear from the context)is a two-sided R-submodule I of Q so that aI ⊆ R and Ib ⊆ R for someinvertible elements a, b ∈ Q. If, furthermore, I ⊆ R then I is called anintegral fractional ideal. If A and B are subsets of Q then we put

(A :r B) = {q ∈ Q | Bq ⊆ A}

and(A :l B) = {q ∈ Q | qB ⊆ A}.

If I is a fractional R-ideal then (I :r I) and (I :l I) are orders in Q that areequivalent to R. We will often use the following characterization to checkwhether a ring is a maximal order.

Theorem 2.5.2. Suppose R is an order in Q. Then R is a maximal orderif and only if R = (I :r I) = (I :l I) for every fractional R-ideal I (orequivalently, for every integral fractional R-ideal).

Assume now that R is a maximal order. It follows that (R :r I) = (R :lI) for any fractional R-ideal I. One denotes this set simply as (R : I) or alsoas I−1. Put I∗ = (R : (R : I)), the divisorial closure of I. If I = I∗ thenI is said to be divisorial . Examples of such ideals are invertible fractionalR-ideals J , that is JJ

′= J

′J = R, for some fractional R-ideal J

′(in this

case, clearly, J′= (R : J)). The following properties are readily verified for

fractional R-ideals I and J ; I ⊆ I∗, I∗∗ = I∗, (IJ)∗ = (I∗J∗)∗ = (I∗J)∗ =(IJ∗)∗, (R : I∗) = (R : I) = (R : I)∗, if I ⊆ J then I∗ ⊆ J∗. The divisorialproduct I ∗ J of two divisorial ideals I and J is defined as (IJ)∗. It turnsout that this product is commutative.

Definition 2.5.3. A prime Goldie ring R is said to be a Krull order if Ris a maximal order that satisfies the ascending chain condition on divisorialintegral ideals.

A prime Goldie ring R is said to be an Asano order if R is a maximalorder with each ideal divisorial, or equivalent, each nonzero ideal of R isinvertible.

Chatters and Jordan ([12]) have defined a unique factorization ring(UFR for short) to be a prime Noetherian ring R in which every nonzero



prime ideal contains a prime ideal P generated by a normal element p, thatis P = Rp = pR. Because of the principal ideal theorem 2.2.2, P is of heightone. If, moreover, R and R/P have no zero divisors for each height oneprime ideal P then R is said to be a unique factorization domain (UFD forshort). The divisor group D(R) of a Krull order R is a free abelian groupwith basis the set of prime divisorial ideals. The latter are primes of heightone and for rings satisfying a polynomial identity the height one primes areprecisely the prime divisorial ideals. Furthermore, the normalizing classgroup of a Krull order, cl(R) is defined as D(R)/P (R), where P (R) is thesubgroup consisting of the principal fractional ideals of R generated by anormal element. A Noetherian unique factorization ring is a Krull orderwith trivial normalizing class group ([12]).

In the next theorem we collect some of the essential properties of PIKrull orders. For details we refer the reader to [9, 10].

Theorem 2.5.4. Let R be a prime Krull order satisfying a polynomialidentity. Then the following properties hold.

1. The divisorial ideals form a free abelian group with basis X1(R), theheight one primes of R.

2. If P ∈ X1(R) then P ∩ Z(R) ∈ X1(Z(R)), and furthermore, for anyideal I of R, I ⊆ P if and only if I ∩ Z(R) ⊆ P ∩ Z(R).

3. R =⋂RZ(R)\P , where the intersection is taken over all height one

primes of R, and every regular element r ∈ R is invertible in almostall (that is, except possibly finitely many) localizations RZ(R)\P . Fur-thermore, each RZ(R)\P is a left and right principal ideal ring with aunique nonzero prime ideal.

4. For a multiplicatively closed set of ideals M of R, the (localized) ringRM = {q ∈ Qcl(R) | Iq ⊆ R, for some I ∈M} is a Krull order, and

RM =⋂RZ(R)\P ,

where the intersection is taken over those height one primes P forwhich RM ⊆ RZ(R)\P .

In the book of Reiner ([67]), the general theory of R-orders can befound. The ring R will always denote a Noetherian integral domain with aquotient field F . A will always be a finite dimensional F -algebra.



Definition 2.5.5. An R-order in the F -algebra A is a subring Λ of A,having the same identity element as A, and such that Λ is a finitely gener-ated module over R and generates A as a linear space over F . An order Λis called maximal, if it is not properly contained in any other R-order.

From Theorem 2.5.4 and other results about PI-rings we get the follow-ing theorem, that gives a nice link between R- orders and the orders weconsidered before.

Theorem 2.5.6. The following conditions on a ring Λ with centre R areequivalent:

1. Λ is a prime PI ring;

2. Λ is an order in a central simple algebra;

3. Λ is an R-order in a central simple algebra.

In particular, if Λ is a PI domain, then Λ is an R-order in a central divisionalgebra.

We will now give a characterization of group algebras of polycyclic-by-finite groups that are Noetherian prime maximal orders. These results aredue to K.A. Brown and can be found in [6, 7]. It is also stated when suchalgebras are unique factorization rings or domains.

Recall that a group G is dihedral free if the normalizer of any subgroupH isomorphic with the infinite dihedral group D∞ = 〈a, b | b2 = 1, ba =a−1b〉 is of infinite index in G, that is, if H ∼= D∞ then H has infinitelymany conjugates in G.

Theorem 2.5.7. Let G be a polycyclic-by-finite group and K a field. Then,K[G] is a Noetherian prime maximal order if and only if ∆+(G) = {1} andG is dihedral free.

Theorem 2.5.8. Let G be a polycyclic-by-finite group and K a field. Then,K[G] is a Noetherian unique factorization ring if and only if ∆+(G) = {1},G is dihedral free and every plinth of G is centric.

From Theorem 2.5.7 it follows that the group algebra of a torsion-freepolycyclic- by-finite group is a maximal order. Furthermore, from Theo-rem 2.5.8 and the following result it follows that the group algebra of afinitely generated torsion-free abelian-by-finite group is a unique factoriza-tion ring.



Theorem 2.5.9. Let G be a polycyclic-by-finite group and K a field. If∆+(G) = {1} then the following conditions are equivalent.

1. Every nonzero ideal of K[G] contains an invertible ideal.

2. Every nonzero ideal of K[G] contains a nonzero central element (thisholds for example if K[G] is a PI algebra).

3. Every nonzero ideal of K[G] contains a nonzero normal element.

4. Every plinth of G is centric.

We end this chapter the way we started it, with semigroups. Definitionsof orders and fractional ideals in the semigroup context are basically thesame as in the ring case (see for example [28] and [71]). A cancellativemonoid S which has a group of left and right quotients G is called anorder. Such a monoid S is called a maximal order if there does not exist asubmonoid S′ of G properly containing S and such that aS′b ⊆ S for somea, b ∈ G. An abelian cancellative monoid S is a maximal order in its groupof quotients G if and only if it is completely integrally closed. The lattermeans that, if s, g ∈ G are such that {sgn | n ∈ N0} ⊆ S then g ∈ S. Incase S is finitely generated, S is completely integrally closed if and onlyif S is integrally closed, that is, if gn ∈ S, with g ∈ G and some positiveinteger n, then g ∈ S.

For subsets A,B of G we define (A :l B) = {g ∈ G | gB ⊆ A} andby (A :r B) = {g ∈ G | Bg ⊆ A}. A f ractional ideal I of S means thatSIS ⊆ I and cI, Id ⊆ S for some c, d ∈ S. If S is a maximal order,then (S :l I) = (S :r I) for any fractional ideal I; we simply denote thisfractional ideal by (S : I) or by I−1. Recall that then I is said to bed ivisorial if I = I∗, where I∗ = (S : (S : I)). A fractional ideal is said tobe invertible if IJ = JI = S for some fractional ideal J of S. In this caseJ = I−1 and I is a divisorial ideal. The divisorial product I ∗ J of twodivisorial ideals I and J is defined as (IJ)∗.

Theorem 2.5.2 yields necessary and sufficient conditions on the idealsfor a ring R to be a maximal order. Furthermore it turns out that, forsemigroups satisfying the ascending chain condition on two-sided ideals,conditions on the prime ideals are enough (see [45, Lemma 8.5.4]).

Lemma 2.5.10. Let S be a cancellative monoid which is an order (in itsgroup of quotients G = SS−1) and suppose S satisfies the ascending chaincondition on two-sided ideals. Then S is a maximal order if and only if(P :l P ) = (P :r P ) = S for all P ∈ Spec(S).



Similar as in the ring case, an order S is said to be a Krull order if andonly if S is a maximal order satisfying the ascending chain condition ondivisorial integral ideals (the latter are the fractional ideals contained in S).In this case, the setD(S) of divisorial fractional ideals is a free abelian groupfor the ∗ operation. If S is a Krull order in a finitely generated abelian-by-finite group (so K[S] satisfies a polynomial identity by Theorem 2.2.5)then the minimal primes of S form a free basis for D(S). Furthermore, thenormalizing class group of a Krull order S, cl(S), is defined as D(S)/P (S),where P (S) is the subgroup consisting of the principal fractional ideals ofS generated by a normal element. In case the semigroup S is abelian wesimply speak about the class group of S.


Chapter 3

Maximal order semigroupalgebras

In this chapter, we give necessary and sufficient conditions for a primeNoetherian algebra K[S] to be a maximal order, provided S is a submonoidof a polycyclic-by-finite group. Before we do this, we give a brief motiva-tion why we focus on the semigroups under consideration and we statesome known results on the topic. All the results in this chapter are jointwork with Eric Jespers and Jan Okninski. The results in Section 3.2 andSection 3.3 have been published in [30] and the results in Section 3.4 willappear in [31].

3.1 Introduction and motivation

As said in Chapter 1, in general, it remains an unsolved problem to charac-terize when an arbitrary semigroup algebra is a prime Noetherian maximalorder. Even for some concrete classes of finitely presented algebras it is noteasily verified when these algebras are maximal orders. We state three veryconcrete examples.

Example 3.1.1. Let K be any field and let R = K〈x1, x2, x3, x4〉 be thealgebra defined by the following relations:

x1x4 = x2x3, x1x3 = x2x4, x3x1 = x4x2

x3x2 = x4x1, x1x2 = x3x4, x2x1 = x4x3.

Is R a prime Noetherian maximal order?

29

3.1. Introduction and motivation 30


x21 = x2

2 = x23 = x2

4

x1x3 = x4x2, x1x4 = x3x2

x2x3 = x4x1, x2x4 = x3x1.



x1x2x3 = x24

x1x2 = x2x1, x1x3 = x3x1, x2x3 = x3x2

x1x4 = x4x2, x2x4 = x4x1, x3x4 = x4x3.


In this thesis constructions are given of Noetherian maximal orders thatare finitely presented algebras over a fieldK, defined by monomial relations.In order to do this, it is shown that the underlying homogeneous informa-tion determines the algebraic structure of the algebra. So, it is naturalto consider such algebras as semigroup algebras K[S] and to investigatethe structure of the monoid S. That is why we will develop techniques tostudy semigroups and semigroup algebras. Then, using our results, we willbe able to solve the questions posed in the three examples stated above.

Clearly, in Example 3.1.1, the algebra R = K[S] where S is the monoiddefined by the same presentation. It turns out that the monoid S is asubmonoid of a finitely generated abelian-by-finite group. In Chapter 4 wecharacterize when such semigroup algebras are prime Noetherian maximalorders (Theorem 4.1.3). As an illustration of this result, we will show thatthe algebra in 3.1.1 is a prime Noetherian maximal order (Example 4.3.2).

Examples 3.1.2 and 3.1.3 are similar. They both concern algebrasR = K[S], where S is a monoid of IG-type and this is a submonoid ofan abelian-by-finite group. Monoids of IG-type are studied in Chapter 6,where we give necessary and sufficient conditions for semigroup algebras ofIG-type to be prime Noetherian maximal orders (Theorem 6.2.13). Fromthis characterization it will follow that the algebra in Example 3.1.2 isa maximal order, while the algebra in Example 3.1.3 is not (see Exam-ples 6.2.14 and 6.2.15).

Maximal order semigroup algebras


Now, let us return to known results on Noetherian maximal orders.Concerning group algebras, the Noetherian and maximal order question hasonly been solved in case one works with polycyclic-by-finite groups. K.A.Brown characterized group algebras K[G] of polycyclic-by-finite groups Gthat are prime Noetherian maximal orders. This turns out to be alwaysthe case if G is a torsion-free polycyclic-by-finite group (see Theorem 2.3.2and Theorems 2.5.7, 2.5.8 and 2.5.9). Hence, when one wants to deal withNoetherian maximal orders in the context of semigroup algebras, it is nat-ural to consider submonoids S of polycyclic-by-finite groups. A structuralcharacterization of such semigroup algebras K[S] that are right Noetherianhas been obtained in [42, 44]. For a survey we refer the reader to [45].Recall that, for a group H, we denote by [H,H] the commutator subgroupof H.

Theorem 3.1.4. Let S be a submonoid of a polycyclic-by-finite group andlet K be a field. Then the following conditions are equivalent.

1. S satisfies the ascending chain condition on right ideals.

2. K[S] is right Noetherian.

3. S has a group of right quotients G with a normal subgroup H of finiteindex such that the commutator subgroup [H,H] ⊆ S and S ∩ H isfinitely generated.

4. S has a group of right quotients G that contains normal subgroups Fand N so that F ⊆ N , F is a subgroup of finite index in the unitgroup U(S) of S, the group N/F is abelian, G/N is finite and S ∩Nis finitely generated. In particular it follows that, if the monoid S hasno units, the group G is abelian-by-finite.

It follows that K[S] is right Noetherian if and only if K[S] is left Noethe-rian. Furthermore, if K[S] is Noetherian, S is finitely generated.

Note that, due to a result of Chin and Quinn ([13]) on rings gradedby polycyclic-by-finite groups, it follows that K[S] is right Noetherian ifand only if S satisfies the ascending chain condition on right ideals. Thisyield the equivalent statements (1) and (2) above. From now on, we willnaturally use this result without mentioning it. The following Lemma istaken from [45, Lemma 4.1.3].

Lemma 3.1.5. Let S be a submonoid of a polycyclic-by-finite group. If thesemigroup algebra K[S] is Noetherian, then K[S ∩ G1] is Noetherian for



every subgroup G1 of G, and, if, furthermore, G1 is of finite index in Gthen K[S] is a finitely generated right (and left) K[S ∩ G1]-module. Also,if S is a submonoid of a finitely generated group G that has an abeliansubgroup A of finite index, then K[S] is Noetherian if and only if S ∩A isfinitely generated.

If S is an abelian monoid, Anderson [1, 2] characterized when K[S]is a prime Noetherian maximal order and, more generally, Chouinard [14]described commutative semigroup algebrasK[S] that are Krull domains. Inparticular, it turns out that the height one prime ideals of K[S] determinedby the minimal primes of S are crucial. In Chapter 5 we go more into detailabout these abelian maximal orders.

On the other hand, if S is a submonoid of a torsion-free finitely gen-erated abelian-by-finite group, Jespers and Okninski described in [41, 45]when K[S] is a Noetherian maximal order that is a domain. The descrip-tion is fully in terms of the monoid S. In Chapter 6 we will give largeclasses of monoids satisfying all the mentioned properties in this theorem.

Theorem 3.1.6. Let K be a field and S a submonoid of a torsion-freefinitely generated abelian-by-finite group. The semigroup algebra K[S] is aNoetherian maximal order if and only if the following conditions are satis-fied:

1. S satisfies the ascending chain condition on one sided ideals,

2. S is a maximal order in its group of quotients G = SS−1,

3. for every minimal prime P in S,

SP = {g ∈ G | Cg ⊆ S for some G-conjugacy class C of G

contained in S and with C * P}

has only one minimal prime ideal.

In this chapter we deal with submonoids S of arbitrary polycyclic-by-finite groups (thus not necessarily torsion-free) and we determine whenK[S] is a prime maximal order. As in the abelian case, knowledge of primeideals is fundamental. Suppose K[S] is Noetherian. Because K[SS−1] isa Noetherian localization of K[S] with respect to the Ore set S, it followsfrom Theorem 2.4.6 that, for any two-sided ideal I of K[S], the extensionIe = IK[G] is a two-sided ideal of K[G]. Furthermore, the map Q 7→ Qe

determines a one-to-one correspondence between the set of prime ideals



Q of K[S] that do not intersect S and the prime ideals of K[G] (sinceQe∩K[S] = Q). The latter are rather well understood through the work ofRoseblade (see for example [64]), and in particular, the height one primescan be handled via Brown’s result. The crucial point in our investigationsare thus the primes of K[S] intersecting S non-trivially.

For a submonoid S of a torsion-free polycyclic-by-finite group G withNoetherian semigroup algebra K[S], the following information was provedin [42]. Recall that G being torsion free is equivalent with K[S] being adomain (see Theorem 2.3.3 and Theorem 2.3.4).

1. If Q is a prime ideal of S, then K[Q] is a prime ideal of K[S].

2. The height one prime ideals of K[S] intersecting S non-trivially areprecisely the ideals K[Q] with Q a minimal prime ideal of S.

In Section 3.2 we continue the study of prime ideals in case K[S] isprime (and thus G is not necessarily torsion free). We show that the sec-ond part on primes of height one still remains true, however, we show inExample 4.3.2 that the first property does not remain valid. As a conse-quence of the study of the prime ideals of K[S], we establish going up andgoing down properties between prime ideals of S and prime ideals of S∩H,where H is a subgroup of finite index in G. These are the analogs of theimportant results (see Theorem 2.2.8) known on the prime ideal behaviorbetween a ring graded by a finite group and its homogeneous componentof degree e (the identity of the grading group). As an application, in Sec-tion 3.3, it is shown that the classical Krull dimension clKdim(K[S]) ofK[S] is the sum of the prime dimension of S and the plinth length of theunit group U(S). Also, a result of Schelter (Theorem 2.2.6) is extended tothe monoid S: the prime dimension of S is the sum of the height and depthof any prime ideal of S.

The information obtained on primes of height one then allows us in Sec-tion 3.4 to determine when a semigroup algebra K[S] is a prime Noetherianmaximal order provided that G = SS−1 is a polycyclic-by-finite group. Theresult reduces the problem to the structure of the monoid S (in particularS has to be a maximal order within its group of quotients G), to that ofG and to the structure of the minimal primes of S. It gives us a solution,under an invariance condition on the minimal primes, to the open questionwhen is a semigroup algebra of a submonoid of a polycyclic-by-finite groupa prime Noetherian maximal order. These results are joint work with EricJespers and Jan Okninski and can be found in the papers [30] and [31].


3.2. Results on height one prime ideals of K[S] 34

3.2 Results on height one prime ideals of K[S]

In Lemma 2.1.8, we stated that a cancellative monoid with no free non-abelian subsemigroups has a group of (two-sided) quotients. In particular,it followed that a submonoid of an abelian-by-finite group has a groupof quotients. In the following lemma, we give a sufficient condition for asubmonoid of a polycyclic-by-finite group to have a group of (two-sided)quotients (see for example [45, Lemma 4.1.9] and [48, Lemma 1.1]).

Lemma 3.2.1. Let S be a submonoid of a polycyclic-by-finite group. If Ssatisfies the ascending chain condition on right ideals, then S has a groupof quotients that is obtained by inverting the normal elements of S (or, byTheorem 3.1.4, by inverting the elements u of S that are central moduloF , where F is a normal subgroup of finite index in the unit group of S).Furthermore, every right ideal of S contains such a normal element u and,for every g, s ∈ S, there exists a positive integer k so that gskg−1 ∈ S.

In particular, if S is a submonoid of a finitely generated abelian-by-finitegroup, the group of quotients is obtained by inverting the central elementsof S and every ideal of S contains a central element.

We now describe the height one prime ideals in prime Noetherian semi-group algebras of submonoids of polycyclic-by-finite groups.

Recall that, for a group H and a set X, MX(H) consists of all X-by-Xmatrices s = (sij) such that each row, and each column, of s contains atmost one nonzero entry and all entries are in H0 = H ∪ {0}. The set ofmatrices with at most one nonzero entry is an ideal of MX(H). We saythat S is a monomial semigroup if S ⊆MX(H) for a group H and a set X.In case |X| = n then we simply denote the monoid MX(H) by Mn(H).For more details we refer the reader to [45].

If S is a monoid with a group of quotients G = SS−1, the semigroupalgebra K[S] is prime if and only if K[G] is prime (Theorem 2.3.4), orequivalently, by Theorem 2.3.3, G does not contain nontrivial finite normalsubgroups.

Theorem 3.2.2. Let S be a submonoid of a polycyclic-by-finite group, saywith a group of quotients G, and let K be a field. Assume that S satisfies theascending chain condition on right ideals and G does not contain nontrivialfinite normal subgroups. Then the height one prime ideals P of K[S] suchthat P ∩ S 6= ∅ are exactly the ideals of the form P = K[Q], where Q is aminimal prime ideal of S.



Proof. Let P be a height one prime ideal of K[S] such that Q = P ∩S 6= ∅.By [62] (or see [45, Theorem 4.5.2], and also [45, Corollary 4.5.7]), S/Qembeds intoMn(H), the semigroup of n×nmonomial matrices over a groupH = TT−1 for a subsemigroup T of S such that T ∩Q = ∅. Furthermore,K[S]/K[Q] embeds in the matrix algebra Mn(K[H]) and the latter is alocalization of K[S]/K[Q] with respect to an Ore set of regular elementsthat does not intersect P/K[Q]. Subsemigroups of S not intersecting Qcan be identified with subsemigroups of Mn(H). It is also known thatthere exists an ideal I of S containing Q such that the nonzero elementsof I/Q are the matrices in S/Q ⊆Mn(H) with exactly one nonzero entry.Furthermore, T may be chosen so that T ⊆ I \ Q and actually T maybe identified with (e11Mn(H)e11 ∩ (S/Q)) \ {0}, where e11 is a diagonalidempotent of rank one in Mn(H). Clearly I/Q is an essential ideal inS/Q. We know that S contains a normal subgroup F of G such that G/Fis abelian-by-finite. Then H ⊆ FH and clearly FH ∩ Q = ∅ and FH =FT (FT )−1. Clearly FTTF ⊆ FHF = FH, whence (FT )(TF ) ∩ Q = ∅.Since T is a diagonal component of I and FT, TF ⊆ I, by the matrixpattern on the matrices of rank one in Mn(H) it follows that TF = FT =T . Hence H = gr(T ) = gr(TF ) = HF .

Notice that we have a natural homomorphism φ : K[S]/K[Q] −→K[S]/P . Since K[S] is Noetherian and as Mn(K[H]) is a localizationof K[S]/K[Q] with respect to an Ore set of regular elements that doesnot intersect P/K[Q], there exists a prime ideal R in K[H] such thatMn(K[H]/R) is a localization of K[S]/P (Theorem 2.4.6). Moreover, sinceP/K[Q] is a prime ideal of K[S]/K[Q], from Theorem 2.4.5, we have thatMn(K[H]/R) ⊆ Mn(Mt(D)) for a division ring D and a positive inte-ger t such that Mt(D) is the ring of quotients of K[H]/R and φ extendsto a homomorphism φ′ : Mn(K[H]) −→ Mn(K[H]/R). Let ρP be thecongruence on S defined by the condition: (x, y) ∈ ρP if x − y ∈ P .Since the image of a nonzero entry of a matrix s ∈ Mn(H) under φ′ isinvertible in K[H]/R, the rank of the matrix φ′(s) is equal to t multi-plied by the number of nonzero entries of s. Thus, the ideal J of ma-trices of rank at most t in S/ρP ⊆ Mnt(D) satisfies φ−1(J) = I. More-over, S/ρP = φ′(S/Q) ⊆ Mn(GLt(D)) inherits the monomial pattern ofS/Q ⊆ Mn(H). From [45, Corollary 4.5.14] we know that there is a rightOre subsemigroup U of I such that h(UU−1) = h(G)−1 and U∩P = ∅, andalso the image U ′ of U in S/ρP is contained in a diagonal component of J(viewed as a monomial semigroup over GLt(D)). Then U can be identifiedwith a subsemigroup of S/Q and it is contained in a diagonal component



of S/Q. It thus follows that h(H) ≥ h(UU−1) = h(G)− 1.By Lemma 3.2.1 we know that every ideal of S contains an element of

S that is central modulo F . So, choose z ∈ Q such that zx ∈ xzF forevery x ∈ S. Suppose that zm ∈ H for some positive integer m. Thenzm = ab−1 for some a, b ∈ T . Hence zmb ∈ T ∩ P . As T ∩ P = ∅, weobtain a contradiction. It thus follows that h(gr(z,H)) > h(H) ≥ h(G)−1.Therefore h(gr(z,H)) = h(G) and, because G is polycyclic-by-finite, we get[G : gr(z,H)] < ∞. Since zs ∈ szF for every s ∈ S ∩ H, it follows thatz(S∩H) ⊆ (S∩H)zF = (S∩HF )z = (S∩H)z. As H = (S∩H)(S∩H)−1,it follows that zH = Hz. Therefore ∆+(H) has finitely many conjugatesin G. Since ∆+(H) is finite, we get that ∆+(H) ⊆ ∆+(G) = {1} and thusK[H] is a prime algebra. Hence, the localization Mn(K[H]) of K[S]/K[Q]is prime. So K[Q] is a prime ideal in K[S]. As K[S] is prime and P is ofheight one, we get that P = K[Q], as desired.

To prove the converse, let Q be a minimal prime ideal of S. Let P bean ideal of K[S] maximal with respect to S ∩P = Q. Clearly P is a primeideal of K[S]. Again by Lemma 3.2.1, we know that there exists an elementz in S that belongs to Q and that is central modulo F . Then zS = Sz,so z is a normal element of K[S]. Since, by assumption K[S] is a primeNoetherian algebra, Theorem 2.2.2 therefore yields a prime ideal P ′ of K[S]that is of height one so that z ∈ P ′ and P ′ ⊆ P . By the first part of theresult, P ′ = K[S ∩ P ′]. Since S ∩ P ′ is a prime ideal of S contained in theminimal prime ideal S ∩ P = Q, we get that S ∩ P ′ = Q. So K[Q] = P ′ isa height one prime ideal of K[S].

We give an easy example that shows that Theorem 3.2.2 can not beextended to semigroup algebras that are not Noetherian.

Example 3.2.3. Let

S = {xiyj | i > 0, j ∈ Z} ∪ {1}

a submonoid of the free abelian group gr(x, y) of rank two. It is easy tosee that for a given i > 0 and j ∈ Z the ideal xiyjS contains the set{xkyj | k > i, j ∈ Z}. Therefore P = S \ {1} is nil modulo the idealxiyjS. It follows that S \ {1} is the only prime ideal of S. Moreoverrk(S/P ) = 0 < rk(S) − 1, while P is a minimal prime ideal of S. ClearlyK[P ] is a prime ideal of K[S]. Let P ′ be the K-linear span of the setconsisting of all elements of the form xi(yj − yk) with i > 0 and j, k ∈ Z.Then P ′ is an ideal of K[S] and K[S]/P ′ is isomorphic to the polynomialalgebra K[x]. Therefore K[P ] is a prime ideal of height two (note that



clKdim(K[S]) = rk(S) = 2, where clKdim stands for the classical Krulldimension (Theorem 2.2.4)).

On the other hand, for Noetherian semigroup algebras one can alsogive an example (see Example 4.3.2) showing that Theorem 3.2.2 cannotbe extended to prime ideals of height exceeding one.

In order to give some applications to the behaviour of prime ideals ofS and those of S ∩H (with H a normal subgroup of finite index in SS−1)we prove the following technical lemma.

Lemma 3.2.4. Let S be a submonoid of a polycyclic-by-finite group, saywith a group of quotients G. Assume that S satisfies the ascending chaincondition on right ideals. Then G has a poly-(infinite cyclic) normal sub-group H of finite index so that S ∩H is G-invariant and H/U(S ∩H) isabelian. If G is abelian-by-finite then H can be chosen to be an abeliansubgroup.

Proof. Recall that G contains a characteristic subgroup C that is poly-(infinite cyclic). Since S satisfies the ascending chain condition on rightideals, it follows from Theorem 3.1.4 that G contains a normal subgroupF so that G/F is abelian-by-finite and F ⊆ S. Hence C ∩ F is a normalsubgroup of G that is poly-(infinite cyclic), C ∩ F ⊆ U(S) and G/(C ∩ F )is abelian-by-finite. It is thus sufficient to prove the result for the monoidS/ ∼C∩F and its group of quotients G/(C ∩ F ). In other words we mayassume that G is abelian-by-finite.

So, let A be a torsion free abelian and normal subgroup of finite indexin G. From Lemma 3.1.5, S ∩A is a finitely generated abelian monoid andits group of quotients is of finite index in G. Since S satisfies the ascendingchain condition on right ideals, we know from Lemma 3.2.1 that for everyg ∈ S and s ∈ S there exists a positive integer k so that gskg−1 ∈ S, andthat G = SZ(S)−1. Hence the latter property holds for all g ∈ G. As S∩Ais finitely generated abelian and G/A is finite, it follows that there exists apositive integer n so that

g(S ∩A)(n)g−1 ⊆ S ∩A,

for all g ∈ G, where by definition (S ∩A)(n) = {sn | s ∈ S ∩A}. Hence

T =⋂g∈G

g(S ∩A)(n)g−1

is a G-invariant submonoid of S. Because A is the group of quotients ofS ∩A, we get that each g(S ∩A)(n)g−1 has a group of quotients that is of



finite index in G. Since there are only finitely many such conjugates, it isclear that TT−1 is of finite index in G. Hence, the result follows.

To end this section we prove the analogs of the important results knownon the prime ideal behavior between a ring graded by a finite group andits homogeneous component of degree e (Theorem 2.2.8).

Corollary 3.2.5. Let S be a submonoid of a polycyclic-by-finite group, saywith a group of quotients G, and let K be a field. Assume that S satisfiesthe ascending chain condition on right ideals. The following properties holdfor a torsion free normal subgroup H of finite index in G.

1. If P is a prime ideal of S then P∩H = Q1∩· · ·∩Qn, where Q1, . . . , Qn

are all the primes of S∩H that are minimal over P∩H. Furthermore,P = M∩S for any prime ideal M of K[S] that is minimal over K[P ],ht(M) = ht(K[Q1]) = · · · = ht(K[Qn]) and P = B(S(Q1∩· · ·∩Qn)S)(the prime radical of S(Q1 ∩ · · · ∩ Qn)S). If, furthermore, H ∩ S isG-invariant then Qi = Qg

1 for some g ∈ G, 1 ≤ i ≤ n.

2. If S ∩H is G-invariant and Q = Q1 is a prime ideal of S ∩H thenthere exists a prime ideal P of S so that P ∩H = Q1∩Q2∩ · · ·∩Qm,where Q1, . . . , Qm are all the prime ideals of S ∩H that are minimalover P ∩H. One says that P lies over Q. Moreover, each Qi = Qgi

for some gi ∈ G.

3. Incomparability Suppose S ∩ H is G-invariant, Q1 and Q2 areprime ideals of S ∩ H, and P1 and P2 are prime ideals of S. IfP1 lies over Q1 and P2 lies over Q2 so that Q1 ⊆ Q2 and P1 ⊆ P2,then P1 = P2 if and only if Q1 = Q2.

4. Going up Assume S∩H is G-invariant. Suppose Q2 is a prime idealof S ∩H and P2 is a prime ideal of S lying over Q2.

(a) If Q1 is a prime ideal of S ∩H containing Q2 then there existsa prime ideal P1 lying over Q1 so that P2 ⊆ P1.

(b) If P1 is a prime ideal of S containing P2 then there exists aprime ideal Q1 of S ∩H containing Q2 so that P1 lies over Q1.

5. Going down Assume S ∩H is G-invariant. Suppose Q1 is a primeideal of S ∩H and P1 is a prime ideal of S lying over Q1.

(a) If Q2 is a prime ideal of S ∩H contained in Q1 then there existsa prime ideal P2 lying over Q2 so that P2 ⊆ P1.



(b) If P2 is a prime ideal of S contained in P1 then there exists aprime ideal Q2 of S∩H contained in Q1 so that P2 lies over Q2.

Proof. The algebra K[S] has a natural gradation by the finite group G/H.Its homogeneous component of degree e (the identity of the group G) isthe semigroup algebra K[S ∩ H]. Let P be a prime ideal of S. Let Mbe a prime ideal of K[S] minimal over K[P ]. Note that then K[S]/K[P ]inherits a natural G/H-gradation, with component of degree e the algebraK[S ∩H]/K[P ∩H]. Because of Theorem 2.2.8 on going-up and down onprime ideals of rings graded by finite groups, one gets that K[S∩H]∩M =P1∩· · ·∩Pn, where P1, . . . , Pn are all the prime ideals of K[S∩H] that areminimal over K[P ]∩K[S ∩H] = K[P ∩H]. Furthermore, ht(Pi) = ht(M)for 1 ≤ i ≤ n. Since H is torsion free, we know that K[Pi ∩H] is a primeideal of K[S ∩ H] (Theorem 3.2.2). Since it clearly contains K[P ∩ H],it follows that Pi = K[Qi], with Qi = Pi ∩ H. Furthermore, since K[Q]is a prime ideal in K[S ∩ H] for every prime ideal Q of S ∩ H, it followsthat Q1, . . . , Qn are all the prime ideals of S ∩ H minimal over P ∩ H.So K[S ∩ H] ∩M = K[Q1 ∩ · · · ∩ Qn]. Because K[S ∩ H]/K[P ∩ H] isNoetherian, we know that its prime radical is nilpotent (Theorem 2.2.1).Hence (Q1 ∩ · · · ∩Qn)k ⊆ P ∩H for some positive integer k. Since H is offinite index in G, it then also follows that M ∩S is an ideal of S that is nilmodulo S(P ∩H)S. Since K[S] is Noetherian, this yields that (M ∩ S)l ⊆S(P ∩H)S ⊆ P , for some positive integer l (see Theorem 2.1.3). As P isa prime ideal, we therefore obtain that M ∩ S = P, P ∩H = Q1 ∩ · · · ∩Qn

and P = B(S(Q1 ∩ · · · ∩Qn)S).Assume now that, furthermore, S ∩H is G-invariant. For an ideal I of

S ∩H put Iinv =⋂

g∈G Ig, the largest invariant ideal of S ∩H contained

in I. Clearly SIinv = IinvS is an ideal of S and SIinv ∩ (S ∩H) = Iinv. Itfollows that

SQinv1 · · ·SQinv

n ⊆ P.

Hence Qinvi ⊆ P ∩ (S ∩H) = Q1 ∩ · · · ∩ Qn for some i. Because S ∩H is

invariant, it follows that every Qgi is a prime ideal of S ∩ H (of the same

height as Qi). Hence, for every 1 ≤ j ≤ n there exists g ∈ G with Qgi = Qj .

This proves the first part of the result.To prove the second part, let Q be a prime ideal of S ∩H and suppose

that S ∩ H is G-invariant. Then, each Qg is a prime ideal of S ∩ H andht(Q) = ht(Qg). Now, if Q′ is a prime ideal of S ∩ H containing Qinv,then Qg ⊆ Q′ for some g ∈ G. Hence, if Q′ is a prime minimal overQinv then Qg = Q′. Clearly, for every h ∈ G, the prime Qh contains


3.3. Applications of primes: Schelter’s theorem for semigroups 40

a prime ideal Q′′ of S ∩ H minimal over Qinv. Hence, by the previous,Q′′ = Qg ⊆ Qh, for some g. Since ht(Qg) = ht(Qh), it thus follows thatQh = Q′′. So, we have shown that the ideals Qg are precisely the primeideals of S ∩H that are minimal over Qinv. Since H is torsion free we thusget that the ideals K[Qg] are precisely the prime ideals of K[S ∩H] thatare minimal over K[Qinv] (Theorem 3.2.2). Since SQinv = QinvS is anideal of S, the algebra K[S]/K[SQinv] has a natural G/H-gradation, withK[S ∩ H]/K[Qinv] as component of degree e. Hence, by Theorem 2.2.8,there exists a prime ideal M of K[S] that is minimal over K[SQinv] andM ∩K[S ∩H] = K[Q] ∩ P2 ∩ · · · ∩ Pm, where K[Q], P2, . . . , Pm are all theprime ideals that are minimal over M ∩K[S ∩H] (and these ideals are ofthe same height as M). Hence each Pi is minimal over K[Qinv] and thus isof the form K[Qg] for some g. It follows that P = M ∩ S is a prime idealof S so that P ∩H is an intersection of Q and some of the prime ideals Qg,with g ∈ G. This proves part two.

Parts (3), (4) and (5) are now immediate consequences of parts (1) and(2) and of the corresponding results on going up and down for rings gradedby finite groups (see Theorem 2.2.8).

3.3 Applications of primes: Schelter’s theoremfor semigroups

A fundamental result (see Theorem 2.3.6) on the group algebra K[G] ofa polycyclic-by-finite group says that clKdim(K[G]) = pl(G). In the fol-lowing result we determine relations between the considered invariants ofS and K[S] for Noetherian semigroup algebras K[S] of submonoids S ofpolycyclic-by-finite groups.

Corollary 3.3.1. Let S be a submonoid of a polycyclic-by-finite group andlet K be a field. Assume that S satisfies the ascending chain condition onright ideals and let G be its group of quotients. Let F be a normal subgroupof G so that F ⊆ S, U(S)/F is finite and G/F is abelian-by-finite. Thefollowing properties hold.

1. dim(S) = rk(G/F ).

2. dim(S) = dim(S ∩ W ) for any normal subgroup W of G of finiteindex.

3. Spec(S) is finite.



4. clKdim(K[S]) = dim(S) + pl(U(S)).

Proof. Because of Lemma 3.2.4, the group G has a poly-(infinite cyclic)normal subgroup H of finite index so that S ∩H is G-invariant and H/Fis abelian for some normal subgroup F of H that is contained in S. Parts(1-5) of Corollary 3.2.5 easily yield that dim(S) = dim(S∩H). It also easilyis verified that dim(S ∩ H) = dim(T ) where T = (S ∩ H)/ ∼F . ClearlyTT−1 is abelian and TT−1 = H/F because [G : H] <∞. Since U(S)/F isfinite, we obtain that U(T ) is finite. Hence dim(T ) = rk(T ) (see [42]). Sopart (1) follows.

To prove part (2) letW be a normal subgroup of finite index inG. Then,S ∩W inherits the assumptions on S and W is the group of quotients ofS ∩W . So, from part (1) we obtain that dim(S) = rk(G/F ) = rk(W/(F ∩W )) = dim(S ∩W ) and thus dim(S) = dim(S ∩W ).

It easily is seen that there is a natural bijective map between Spec(S)and Spec(S/ ∼F ). Hence to prove part (3) we may assume that SS−1 isabelian-by-finite. Because of Lemma 3.2.4 and part (1) of Corollary 3.2.5,we obtain that Spec(S) is finite if the corresponding property holds forfinitely generated abelian monoids A = 〈a1, . . . , an〉. This is obviouslysatisfied. Indeed, if P ∈ Spec(A) then A \ P is a submonoid of A andai ∈ P for some i. It follows that A \ P is generated by a proper subset of{a1, . . . , an}, whence |Spec(A)| <∞.

Finally, we prove part (4). From [45, Corollary 4.5.14] we know that

clKdim(K[S]) = rk(G/F ) + pl(U(S)).

So the statement follows at once from part (1).

In the following proposition it is shown that the prime spectra of S andS/ ∼∆+(SS−1) can be identified.

Proposition 3.3.2. Let S be a submonoid of a polycyclic-by-finite groupG = SS−1 and assume that S satisfies the ascending chain condition onright ideals. Let H = ∆+(G), the maximal finite normal subgroup of G.Let ∼ be the congruence relation on S determined by H, that is, s ∼ t ifand only if sH = tH. Then the map Spec(S) −→ Spec(S/∼), defined byP 7→ P/∼, is a bijection.

Proof. Let ϕ : S −→ S/∼ ⊆ G/H be the natural epimorphism. Let P bea prime ideal of S.



We claim that P = ϕ−1(ϕ(P )). One inclusion is obvious. To prove theconverse, let x ∈ ϕ−1(ϕ(P )). Then there exists p ∈ P such that x ∼ p. Sox = ph for some h ∈ H. Let s ∈ S. Then

(hsp)n = hhsph(sp)2 · · ·h(sp)n−1(sp)n,

for every n ≥ 1. Because h ∈ H, there exists a positive integer k so thath(sp)k

= h. Define n = (k − 1)|H|+ 1. Then

(hsp)n = (hhsph(sp)2 · · ·h(sp)k−1)|H|(sp)n = (sp)n ∈ P.

It follows that xS is nil modulo P . Because K[S] is Noetherian, fromTheorem 2.1.3 we get that x ∈ P . This proves the claim.

The claim easily implies that ϕ(P ) is a prime ideal of S and the state-ment follows.

We can now prove for the semigroups under consideration an analogueof Schelter’s theorem on prime affine algebras that satisfy a polynomialidentity (Theorem 2.2.6).

Proposition 3.3.3. Let S be a submonoid of a polycyclic-by-finite group,say with a group of quotients G. Assume that S satisfies the ascending chaincondition on right ideals. Let P be a prime ideal of S. Then dim(S/P ) +ht(P ) = dim(S). Furthermore, if U(S) is finite, then dim(S/P ) = rk(S/P ).

Proof. Again let F be a normal subgroup of G so that F ⊆ U(S), [U(S) :F ] <∞, S ∩ F is finitely generated and G/F is abelian-by-finite. Becauseof the natural bijection between Spec(S) and Spec(S/ ∼F ), we may replaceS by S/ ∼F and thus we may assume that G is abelian-by-finite and U(S)is finite. Hence, by Lemma 3.2.4, G contains a normal torsion free abeliansubgroup A so that [G : A] < ∞, T = S ∩ A is finitely generated andG-invariant.

We now first prove that dim(T/Q)+ht(Q) = dim(T ) for a prime ideal Qin the abelian monoid T . Since A is torsion free, we know (Theorem 3.2.2)that K[Q] is a prime ideal of K[T ] (which is of height one if Q is a minimalprime of T ). Hence (by Schelter’s result for finitely generated commutativealgebras)

clKdim(K[T ]) = clKdim(K[T ]/K[Q]) + ht(K[Q]).

Note that T\Q is a submonoid ofA and clKdim(K[T ]/K[Q]) = clKdim(K[T\Q]). Consequently, by Corollary 3.3.1, dim(T ) = dim(T \ Q) + ht(K[Q]).



Since dim(T\Q) = dim(T/Q), we need to prove that ht(K[Q]) = ht(Q). Weprove this by induction on ht(Q) (note that by Corollary 3.3.1 the primespectrum of T is finite and hence every prime contains a minimal primeideal of T ). If ht(Q) = 1 then the statement holds. So, assume ht(Q) > 1.Let Q1 be a prime of height one contained in Q. Then, by the induction hy-pothesis, ht(Q/Q1) = ht(K[Q/Q1]). Since K[(T/Q1)/(Q/Q1)] ∼= K[T/Q],we thus get from Schelter’s result that

clKdim(K[T ])− 1 = clKdim(K[T/Q1])= clKdim(K[T/Q]) + ht(Q/Q1)= clKdim(K[T ])− ht(K[Q]) + ht(Q/Q1).

Hence ht(K[Q]) = ht(Q/Q1) + 1 ≤ ht(Q). Since K[M ] is prime in K[T ] ifM is prime in T , it is clear that ht(Q) ≤ ht(K[Q]). Hence we obtain thatht(K[Q]) = ht(Q), as desired.

Now let P be a prime ideal of S. Because of Corollary 3.2.5, dim(S) =dim(S∩A), dim(S/P ) = dim(T/Q) and ht(P ) = ht(Q), where Q is a primeideal of T and P lies over Q. From the previous it thus follows that

dim(S/P ) + ht(P ) = dim(S).

So, only the last part of the statement of the result remains to be proven.Let Q1 = Q and write P ∩ T = Q1 ∩ · · · ∩ Qn, where Q1, . . . , Qn are allprimes minimal over P ∩ T . We know that clKdim(K[T/Qi]) = rk(T/Qi).Furthermore, because TT−1 is torsion free and K[T ] is Noetherian, we alsoknow that K[Qi] is a prime ideal of K[T ] that is minimal over K[P ∩ T ].Hence, K[Qi/(P ∩ T )] is a minimal prime ideal in the finitely generatedcommutative algebra K[T/(P ∩ T )]. It follows that

clKdim(K[T/(P ∩ T )]) = clKdim(K[T/Qi]) + ht(K[Qi/(P ∩ T )])= clKdim(K[T/Qi]) = rk(T/Qi). (3.1)

Since K[T/(P ∩ T )] is a finitely generated commutative algebra, we knowfrom Theorem 2.2.4 that clKdim(K[T/(P ∩T )]) = GK(K[T/(P ∩T )]) andthat

rk(T/(P ∩ T )) = clKdim(K[T/(P ∩ T )]).

Using (3.1) we thus get

rk(T/(P ∩ T )) = clKdim(K[T ]/(K[Q1] ∩ · · · ∩K[Qn]))= sup{clKdim(K[T/Qi]) | 1 ≤ i ≤ n}= clKdim(K[T/Q]) = rk(T/Q).


3.4. Algebras of submonoids of polycyclic-by-finite groups 44

Since TT−1 is of finite index in SS−1, it follows that

rk(S/P ) = rk(T/(T ∩ P )) = rk(T/Q).

Because U(T/Q) is finite, Corollary 3.3.1 yields that rk(T/Q) = dim(T/Q)and thus we get that

rk(S/P ) = rk(T/Q) = dim(T/Q) = dim(T )− ht(Q)= dim(S)− ht(P ) = dim(S/P ).

This finishes the proof.

3.4 Algebras of submonoids ofpolycyclic-by-finite groups

Let S be a submonoid of a polycyclic-by-finite group such that the semi-group algebra K[S] is Noetherian. Hence, S has a group of quotientsG = SS−1 with normal subgroups F and N such that F ⊆ S ∩ N , N/Fis abelian, G/N is finite and S ∩ N is finitely generated (Theorem 3.1.4).Without loss of generality we may assume that the groups N and N/F aretorsion free. Because of the natural bijection between the prime ideals ofS and the prime ideals of S/F , with the use of Lemma 2.5.10, it is easilyverified that S is a maximal order in its group of quotients G if and only ifthe semigroup S/F is a maximal order in its group of quotients G/F . Thefollowing lemma is well known and straightforward to prove.

Lemma 3.4.1. Let S be a submonoid of a polycyclic-by-finite group suchthat K[S] is Noetherian. If K[S] is a maximal order then S is a maximalorder.

Proof. Assume K[S] is a maximal order and G is the group of quotients ofS. Let I be an ideal of S and assume g ∈ G is such that gI ⊆ I. ClearlyK[I] is an ideal of K[S] and g ∈ (K[I] :l K[I]). Since, by assumption K[S]is maximal order, it follows from Theorem 2.5.2 that g ∈ K[S] ∩ G = S.Similarly (I :r I) = S. Thus S is a maximal order.

Recall from Lemma 3.1.5 that also K[S ∩N ] is Noetherian. MoreoverK[S ∩ N ] is a domain as N is torsion free (Theorem 2.3.4). In order toprove the main theorem we need the following proposition.



Proposition 3.4.2. Let S be a submonoid of a polycyclic-by-finite groupG such that the semigroup algebra K[S] is Noetherian. Then the semigroupS ∩N is a maximal order in its group of quotients if and only if the semi-group algebra K[S ∩N ] is a maximal order in its classical ring of quotients(with N a torsion free subgroup of finite index in G = SS−1, as above).

Proof. If K[S ∩N ] is a prime maximal order, it follows as in Lemma 3.4.1,that the semigroup S ∩ N is a maximal order. Conversely, assume S ∩ Nis a maximal order in its group of quotients N . Because N is torsion free,we know that K[S ∩N ] is a Noetherian domain. To prove that K[S ∩N ]is a maximal order, let I be a non-zero ideal of K[S ∩ N ] and let 0 6=q ∈ Qcl(K[S ∩ N ]) be such that qI ⊆ I. Then qIK[N ] ⊆ IK[N ]. Notethat IK[N ] is a two-sided ideal of K[N ] (see Theorem 2.4.6). Because Nis a torsion free polycyclic-by-finite group, we know from Brown’s result(Theorem 2.5.7) that K[N ] is a maximal order. Hence, it follows thatq ∈ K[N ]. Now, since K[N ] = K[F ] ∗ (N/F ), a crossed product of thefinitely generated torsion free abelian group N/F over the group algebraK[F ], we have that q =

∑ni=1 αiqi, with αi ∈ K[F ] and all qi are in a

transversal of F in N . The image of qi in N/F we denote by qi. Let ≤denote an ordering on the free abelian group N/F . Then, we may assumethat q1 < · · · < qn. Every nonzero element of I can be written in the formβt+α for some t ∈ S ∩N, 0 6= β ∈ K[F ] and some α ∈ K[S ∩N ] such thats < t for all s ∈ supp(α) (if α 6= 0). Let h(I) denote the set consisting ofall such possible elements t ∈ S ∩N . Then h(I) is an ideal of S ∩N . SinceqI ⊆ I, N/F is ordered and K[N ] is a domain, we get (using a standardgraded algebra argument) that qn(h(I)) ⊆ h(I). As S ∩ N is a maximalorder, this implies that qn ∈ S∩N and thus αnqn ∈ (I :l I)∩K[S∩N ]. So,q−αnqn ∈ (I :l I) and |supp(q−αnqn)| < |supp(q)|. Hence, by an inductionargument, we may assume that q− αnqn ∈ K[S ∩N ]. So q ∈ K[S ∩N ], asdesired.

Similarly, one shows that (I :r I) = K[S ∩N ].

A group is residually finite if and only if the intersection of all its normalsubgroups of finite index is trivial.

Theorem 3.4.3. Let S be a submonoid of a polycyclic-by-finite group suchthat the semigroup algebra K[S] is Noetherian, i.e., there exist normal sub-groups F and N of G = SS−1 such that F ⊆ S ∩N , N/F is abelian, G/Nis finite and S ∩ N is finitely generated. Suppose that for every minimalprime P of S the intersection P ∩N is G-invariant.



Then, the semigroup algebra K[S] is a prime maximal order if and onlyif the monoid S is a maximal order in its group of quotients G, the groupG is dihedral-free and ∆+(G) = {1}.

Proof. First note that the G-invariance of P ∩N , for every minimal primeideal P of S, is inherited on P ∩ M = (P ∩ N) ∩ M , for any normalsubgroup M of G with M ⊆ N and N/M finite. In particular, we may forthe remainder assume that N is torsion free (and N/F is torsion free).

Assume K[S] is a prime maximal order. From Lemma 3.4.1, it followsthat S is a maximal order in G. Because of Lemma 3.2.1, G is obtainedfrom S by localizing at a set of normal elements. Hence K[G] is a local-ization of K[S] with respect to an Ore set of normal elements, and thus,by Theorem 2.5.4, K[G] is a prime maximal order as well. By Brown’sresult (Theorem 2.5.7), it follows that the the group G is dihedral-free and∆+(G) = {1}.

For the converse implication, suppose that S is a maximal order in itsgroup of quotients G, the group G is dihedral-free and ∆+(G) = {1}. ByTheorem 2.3.3 and Theorem 2.3.4, K[G] and K[S] are prime. Again, byBrown’s result, K[G] is a maximal order.

Let P be a minimal prime ideal of S. Then, by Theorem 3.2.2, K[P ]is a height one prime of K[S]. Clearly, K[S] has a natural G/N -gradationwith homogeneous component of degree e (the identity of G/N) the algebraK[S ∩N ]. So, from Theorem 2.2.8, it then follows that

P (N) = K[P ] ∩K[S ∩N ] = K[P ∩N ] = Q1 ∩ · · · ∩Qn,

with each Qi a height one prime ideal of K[S ∩ N ]; and these are all theheight one primes of K[S ∩ N ] containing P (N). By Lemma 3.2.4, wecan assume that S ∩ N is G-invariant. Hence because of the assumptionon the invariance of P ∩ N , it is easily verified that the set {Q1, . . . , Qn}is a full orbit (under the conjugation action of G) of height one primesin K[S ∩ N ]. Clearly Qi ∩ (S ∩ N) 6= ∅. So, again by Theorem 3.2.2,Qi = K[Qi ∩ (S ∩N)]; moreover, Qi ∩ (S ∩N) is a minimal prime ideal ofS ∩ N and these are all the minimal primes of S ∩ N containing P ∩ N .Because K[S ∩N ] is Noetherian and N is a polycyclic-by-finite group, weknow (again from Lemma 3.1.5) that each Qi contains a normal elementni, that is an element such that (S ∩ N)ni = ni(S ∩ N). Furthermore,because S is a maximal order, we get that S/F is a maximal order inits group of quotients G/F , in which N/F is abelian and of finite index.Hence, it follows that (S ∩N)/F is a maximal order as well. Consequently,



S ∩ N is a maximal order. So, by Proposition 3.4.2, the Noetherian al-gebra K[S ∩ N ] is a maximal order. Since each Qi is a height one primecontaining a divisorial ideal (namely K[(S ∩ N)ni]), it therefore followsthat it is a divisorial height one prime ideal. It then follows from Propo-sition 1.9 and Proposition 1.10 in [10] that each Qi is localizable (the lo-calization will be denoted K[S ∩ N ]Qi) and hence also that P (N) is alocalizable semiprime ideal of K[S ∩ N ]. Furthermore, K[S ∩ N ] eP (N)

=K[S ∩ N ]Q1 ∩ · · · ∩ K[S ∩ N ]Qn . Here we denote by K[S ∩ N ] eP (N)

thelocalization of K[S ∩ N ] with respect to the set CK[N ](P ) = {c ∈ K[S ∩N ] | c + K[P ∩ N ] is a regular element of the ring K[S ∩ N ]/K[P ∩ N ]}.Consider the G/N -gradation on K[S]. Moreover (see for example [64,Lemma 13.3.5]), CK[N ](P ) is an Ore set of regular elements ofK[S] and thusan element c ∈ K[S∩N ] belongs to CK[N ](P ) if and only if c+K[P ] is regu-lar inK[S]/K[P ]. We begin by showing that the localized ringK[S]CK[N ](P )

is a maximal order. To do so, we show that K[S]CK[N ](P ) is a local ringwith unique maximal ideal PK[S]CK[N ](P ) and so that PK[S]CK[N ](P ) isinvertible and every proper non-zero ideal of K[S]CK[N ](P ) is of the form(PK[S]CK[N ](P ))k for some positive integer k.

From [9, Theoreme 4.1.6] we know that K[S ∩ N ] eP (N)is a semi-local

maximal order that is a principal left and right ideal ring. In particular, itis an Asano order, it has dimension one and its Jacobson radical is equalto P (N)K[S ∩ N ] eP (N)

. Since this ring is the component of degree e ofthe G/N -graded ring K[S]CK[N ](P ), it follows, from Theorem 2.2.8, thatK[S]CK[N ](P ) also has dimension one. By the above, it follows that thenon-zero prime ideals of K[S ∩ N ] eP (N)

are precisely the n prime idealsQiK[S ∩N ] eP (N)

= K[Qi ∩ S ∩N ]K[S ∩N ] eP (N). Consequently, the height

one primes of K[S ∩ N ] eP (N)are of the form K[S ∩ N ] eP (N)

I, where I isan ideal of S ∩N . Since K[S ∩N ] eP (N)

is an Asano order, all its non-zeroideals are products of height one prime ideals. Hence all non-zero idealsof K[S ∩ N ] eP (N)

are of the form K[S ∩ N ] eP (N)I, where I is an ideal of

S ∩ N . We now show that PK[S]CK[N ](P ) is the only height one prime ofK[S]CK[N ](P ). So, let Q be a height one prime ideal of K[S]CK[N ](P ). Again,because of the G/N -gradation and Theorem 2.2.8, I = Q∩K[S∩N ] eP (N)

=K[S ∩N ∩Q] eP (N)

is a non-zero semiprime ideal of K[S ∩N ] eP (N). Hence,

as explained above, non-zero ideals of K[S ∩N ] eP (N)are generated by their

intersection with S∩N , and (Q∩S)∩N = I ∩ (S∩N) is an intersection ofsome of theQi∩(S∩N). In particular, Q∩S 6= ∅. Because of Theorem 2.2.8,



Q ∩K[S] is a height one prime ideal of K[S] and thus, by Theorem 3.2.2Q ∩K[S] = K[Q ∩ S] and Q ∩ S is a minimal prime ideal of S.

The assumptions therefore imply that Q∩ (S∩N) is G-invariant. SinceQ ∩ (S ∩N) ⊆ Qi, for some i, we thus obtain that

Q ∩ (S ∩N) ⊆n⋂

i=1

Qi = K[P ∩N ].

From Corollary 3.2.5, it follows that

P = B(S((Q1 ∩ (S ∩N)) ∩ · · · ∩ (Qn ∩ (S ∩N)))S) = Q ∩ S,

where B(J) denotes the prime radical of an ideal J of S. It follows thatQ = PK[S]CK[N ](P ), as desired.

As, by assumption, S is a maximal order, it follows that P (S : P )is an ideal of S that is not contained in P . Hence PK[S]CK[N ](P )(S :P ) is an ideal of K[S]CK[N ](P ) that is not contained in PK[S]CK[N ](P ).Consequently, PK[S]CK[N ](P )(S : P ) = K[S]CK[N ](P ), i.e. PK[S]CK[N ](P )

is invertible. In particular, by [58, Proposition 4.2.6] this ideal satisfiesthe Artin-Rees property. So, by a result of P. Smith (see [64, Theorem11.2.13]),

⋂k(PK[S]CK[N ](P ))k = {0}. It then easily follows (and it is well

known) that every proper non-zero ideal of K[S]CK[N ](P ) is of the form(PK[S]CK[N ](P ))k, for some unique positive integer k. So, each non-zeroideal of K[S]CK[N ](P ) is invertible. This proves the desired properties ofK[S]CK[N ](P ) and thus K[S]CK[N ](P ) is a maximal order.

Next we will prove that⋂

P∈X1(S)K[S]CK[N ](P ) ∩ G = S. For this,suppose g ∈

⋂P∈X1(S)K[S]CK[N ](P ) ∩ G. Then, for every minimal prime

P of S, there exists an element βP ∈ CK[N ](P ) such that βP g ∈ K[S].We can assume that the image βP of βP is central in K[G/F ]. Indeed,since P (N) is G-invariant, it follows that

∏g∈T β

gP ∈ CK[N ](P ) (product

in any fixed order) for some finite transversal T for N in G. Since N/F isabelian, it follows that

∏g∈T β

gP is central in K[G/F ] and we can replace

βP by this product. Furthermore, βPSg ⊆ K[S/F ] = K[S] and henceSsupp(βP )Sg ⊆ S. The union

⋃P Ssupp(βP )S, over all the minimal primes

P of S, is an ideal of S that is not contained in any minimal prime P ofS. Since also

⋃P Ssupp(βP )Sg ⊆ S, and, since S is a maximal order, it

follows that g ∈ S. Hence g ∈ S, as desired.Because of the remark stated in the beginning of the proof, the previous

holds for any normal subgroup M of G with M ⊆ N and N/M finite.


3.5. Comments 49

Finally, we prove the following claim:

K[S] =⋂

P∈X1(S),M

K[S]CK[M ](P ) ∩K[G],

with M running through all torsion free normal subgroups of G withM ⊆ N and N/M finite. Note that this claim implies the result, i.e.,K[S] is a maximal order. Indeed, let I be a non-zero ideal of K[S] andsuppose that q ∈ Qcl(K[S]) is such that qI ⊆ I. Let P be a minimalprime of S and let M be a subgroup as described. Since qI ⊆ I, we getqIK[S]CK[M ](P ) ⊆ IK[S]CK[M ](P ), with IK[S]CK[M ](P ) a two-sided ideal ofK[S]CK[M ](P ) by Theorem 2.4.6. As K[S]CK[M ](P ) is a maximal order, thisyields q ∈ K[S]CK[M ](P ). On the other hand, as also qIK[G] ⊆ IK[G] andas K[G] is a maximal order, we get that q ∈ K[G]. Hence the claim im-plies that q ∈ K[S]. So we have shown that (I :l I) = K[S]. Similarly,(I :r I) = K[S], and thus indeed K[S] is a maximal order.

So, to prove the claim, let q =∑n

i=1 kigi ∈⋂

P∈X1(S),M K[S]CK[M ](P ) ∩K[G], where ki 6= 0 ∈ K and gi ∈ G for each 1 ≤ i ≤ n and gi 6= gj fori 6= j. It is enough to show that q ∈ K[S]. We prove this by inductionon n. If n = 1 then q = kg with g ∈

⋂P∈X1(S),M K[S]CK[M ](P ) ∩ G and it

follows from the above that g ∈ S, as desired. Hence assume n > 1.Because G is residually finite, there exists a normal subgroup of finite

index M0 in G such that M0 ⊆ N and gig−1j 6∈ M0 for all i 6= j. Note

that CM1∩M2(P ) ⊆ CM1(P ) ⊆ CK[N ](P ) for any two normal subgroupsM1,M2 of G so that M1,M2 ⊆ N and each N/Mi is finite. Hence, in theintersection

⋂P∈X1(S),M K[S]CK[M ](P ) ∩K[G] we may assume that M runs

through all normal subgroups of G with M ⊆ M0 and M0/M finite. Inother words we may replace N by M0 in the intersection. It follows thatthe intersection

⋂P∈X1(S),M K[S]CK[M ](P ) ∩ K[G] is a G/M0-graded ring.

Hence, the induction hypothesis yields that we may assume that q is G/M0-homogeneous, that is, each gig

−1j ∈ M0. Consequently, n = 1 and thus by

the above we get q ∈ K[S]. This ends the proof.

3.5 Comments

Suppose that in Theorem 3.4.3 one also assumes that the group G = SS−1

is abelian-by-finite. Then, in the next chapter, it is shown that the condi-tion “for every minimal prime P of S the intersection P ∩N is G-invariant”is necessary for K[S] to be a maximal order. It is unknown whether this ne-cessity holds in general, nor it is known whether this condition is redundant.


3.5. Comments 50

That is, no example of a maximal order S with the ascending chain con-dition on right ideals in a polycyclic-by-finite group G (with ∆+(G) = {1}and G dihedral-free) is known so that K[S] is not a maximal order. Propo-sition 3.4.2 shows that if such a monoid S exists then G does not containa normal subgroup F so that F ⊆ U(S) and G/F is torsion free abelian.


Chapter 4

Maximal order semigroupalgebras satisfying apolynomial identity

In this chapter we describe when a semigroup algebra K[S] of a cancellativesubmonoid S of a polycyclic-by-finite group G is a prime Noetherian maxi-mal order that satisfies a polynomial identity. So, because of Theorem 2.2.5,we assume that G = SS−1 is a finitely generated abelian-by-finite group.The result, which relies on the information obtained on primes of heightone (Theorem 3.2.2), reduces the problem to the structure of the monoid S.Recall that, in Theorem 3.4.3, we characterized semigroup algebras K[S] ofsubmonoids of polycyclic-by-finite groups that are maximal orders, underan invariance condition on the minimal primes of S.

In Section 4.1, we show that this invariance condition is also necessary,provided K[S] satisfies a polynomial identity. In Section 4.2, we prove auseful criterion for verifying the maximal order property of a submonoidS of a finitely generated abelian-by-finite group. We then show how this,together with our main result (Theorem 4.1.3), can be used to build newexamples of finitely presented algebras R (defined by monomial relations)that are maximal orders (Example 4.3.2). Since the main result of thechapter deals with semigroup algebras K[S] of submonoids S of groups, wefirst have to check that R is defined by such submonoids. So, in particular,we show how to go from the language of presentations of algebras to thelanguage of monoids and their semigroup algebras. The results in thischapter have appeared in [30] and are joint work with Eric Jespers and JanOkninski.

51

4.1. Algebras of submonoids of abelian-by-finite groups 52

4.1 Algebras of submonoids of abelian-by-finitegroups

In this section we characterize when a semigroup algebra K[S] of a sub-monoid S of a finitely generated abelian-by-finite groupG is a prime Noethe-rian maximal order that satisfies a polynomial identity. In case G is torsionfree our result is equivalent to Theorem 3.1.6. First we need the next lemma([45, Lemma 7.1.1]) that yields some consequences on S ∩ A when S is amaximal order. For the convenience of the reader we include a short proof.

Lemma 4.1.1. Let S be a submonoid of an abelian-by-finite group G =SS−1 and let A be an abelian normal subgroup of finite index in G.

1. If S is a maximal order, then S ∩A is a maximal order.

2. If S ∩A is a maximal order, then S ∩A is G-invariant.

Proof. To prove the first part, suppose S is a maximal order. Let I bean ideal of S ∩ A and let a ∈ A = gr(S ∩ A) be such that aI ⊆ I. ThenaJ ⊆ J with J =

∏f∈F f

−1If , where F is a transversal for A in G. Notethat J ⊆ A because A is normal in G. By Lemma 3.2.1, G = SZ(S)−1 =S(Z(S) ∩ A)−1 and because F is finite, there exists b ∈ Z(S) ∩ A withbJ ⊆ S ∩A. Hence J is a fractional ideal of S ∩A. Furthermore SJ = JS,because g−1Jg = J for all g ∈ G. Also bJS ⊆ S and thus JS is a fractionalideal of S. As aJS ⊆ JS and S is a maximal order, one obtains thata ∈ S ∩A, as desired.

For the second part, suppose S ∩ A is a maximal order. Let g ∈ G.Again by Lemma 3.2.1, G = SZ(S)−1 and hence there exists z ∈ Z(S)∩Aso that zg, zg−1 ∈ S. Consider the monoid M = (S ∩ A)g−1(S ∩ A)g.Clearly S ∩A ⊆M ⊆ A and z2M ⊆ S ∩A. Because, by assumption, S ∩Ais a maximal order, it follows that M = S ∩A. Hence g−1(S ∩A)g ⊆ S ∩Afor any g ∈ G. So indeed S ∩A is G-invariant.

We now prove some necessary condition for K[S] to be a prime Noethe-rian maximal order that satisfies a polynomial identity.

Lemma 4.1.2. Let S be a submonoid of an abelian-by-finite group G =SS−1 and let K be a field. Let A be an abelian normal subgroup of finiteindex in G and let P be a minimal prime ideal of S.

If K[S] is a prime Noetherian maximal order, then S is a maximal orderand P ∩A is G-invariant.

Maximal order semigroup algebras satisfying a polynomial identity

4.1. Algebras of submonoids of abelian-by-finite groups 53

Proof. From Lemma 3.4.1, it follows that S is a maximal order. Becauseof Lemma 4.1.1, we know that S ∩ A is G-invariant. Let P be a minimalprime ideal of S. Of course, A ⊆ ∆(G) and thus

P ∩A = (P ∩∆(G)) ∩ (A ∩ S).

Hence to prove that P ∩A is G-invariant, we may assume in the remainderthat A = ∆(G). Indeed, since K[S] is prime, by Theorem 2.3.4, K[G] isprime. Hence, by Theorem 2.3.3, ∆+(G) = {1} and thus ∆(G) is torsionfree abelian. Now, by Corollary 3.2.5, P ∩ A = Q1 ∩ · · · ∩Qn, an intersec-tion of minimal primes of A that are G-conjugate. We need to show that{Q1, . . . , Qn} = {Qg

1 | g ∈ G}. Suppose the contrary, so assume Q′ is aminimal prime of S ∩A that is conjugate to Q1 but is different from all Qi,for 1 ≤ i ≤ n. Then, by Corollary 3.2.5, there exists a prime ideal P ′ of Sso that P ′ ∩ A = Q′ ∩Q′2 ∩ · · · ∩Q′m for some minimal primes Q′2, . . . , Q

′m

of S ∩ A. Because of Theorem 3.2.2, both K[P ] and K[P ′] are distinctheight one prime ideals of K[S]. As, by assumption, K[S] is a prime PINoetherian maximal order, it follows from Theorem 2.5.4 that there existsa central element of K[S] that belongs to K[P ′] but not to K[P ]. Sincecentral elements of K[S] are linear combinations of finite conjugacy classsums of K[G] we get that P ′ contains a G-conjugacy class C that does notbelong to P . Since A = ∆(G), we thus get that C ⊆ A. Hence, C ⊆ Q′.As Q′ is a G-conjugate of Q1 it follows that C ⊆ Q1 ∩ · · · ∩Qn = P ∩A, acontradiction.

From this lemma the main theorem of this chapter follows. Recall thatthe sufficiency of the conditions follows immediately from the polycyclic-by-finite case (Theorem 3.4.3), although the necessity is only proved in thecase that S is a submonoid of a finitely generated abelian-by-finite group.

Theorem 4.1.3. Let K be a field and let S be a submonoid of a finitelygenerated abelian-by-finite group. Let A be an abelian normal subgroup offinite index in G = SS−1. The following conditions are equivalent.

1. K[S] is a prime Noetherian maximal order.

2. S is a maximal order that satisfies the ascending chain condition onright ideals, ∆+(G) = {1}, G is dihedral free and for every minimalprime ideal P of S the set P ∩A is G-invariant.

Proof. Assume that K[S] is a prime Noetherian maximal order. Then,by Theorem 2.5.4, the localization K[G] = K[SZ(S)−1] of K[S] also is a


4.2. Constructing examples of maximal orders 54

maximal order. Hence, again by Theorem 2.5.7, ∆+(G) = {1} and G isdihedral free. Lemma 4.1.2 then yields that the other conditions listed in(2) hold as well.

Conversely, assume that condition (2) holds. The assumption on ∆+(G)yields that K[G] and thus K[S] is prime. Note that, if we assume inTheorem 3.4.3 that the polycyclic-by-finite group is also abelian-by-finite, itfollows that the subgroup N in the theorem is abelian. Hence, the conditionP ∩ A is G-invariant implies that P ∩ N is G-invariant, and the resulttherefore follows from Theorem 3.4.3.

4.2 Constructing examples of maximal orders

Theorem 4.1.3 reduces the problem of determining when K[S] is a primeNoetherian maximal order to the algebraic structure of S. It hence pro-vides a strong tool for constructing new classes of such algebras. For someexamples the required conditions on S can easily be verified, but on theother hand, for some examples this still requires substantial work. In thissection this is illustrated with some concrete constructions.

A first class of examples consists of algebras defined by monoids of I-type (see Chapter 6 and [27, 45]). In particular, these algebras are quadraticalgebras R with a presentation defined by n generators x1, . . . , xn and with(n2

)relations of the form xixj = xkxl so that every word xixj appears at

most once in one of the defining relations. Clearly R = K[S], where S isthe monoid defined by the same presentation. It turns out that S has agroup of quotients G that is torsion free and has a free abelian subgroupA of finite index. The only noncommutative algebra of such type whichis generated by two elements is K〈x, y | x2 = y2〉 (see Example 6.3.3). InChapter 6, using Theorem 4.1.3, we will show that such algebras R areNoetherian maximal orders and domains.

In Chapter 3, we introduced three other examples of finitely presentedalgebras. These examples are not defined by monoids of I-type, but Ex-ample 3.1.2 and 3.1.3 yield examples of some larger class of monoids, theso-called monoids of IG-type. In Theorem 6.2.13, we will prove a charac-terization, for such semigroup algebras, to be a maximal order and, as anillustration of this theorem, we will deal with these two examples.

In the next section we discuss in full detail one more construction thatillustrates Theorem 4.1.3. This example is our earlier mentioned Exam-ple 3.1.1 and it shows that certain assumptions in Theorem 3.2.2 are es-sential. But, before doing this, we establish a useful general method for


4.3. An illustration of the main theorem 55

constructing nonabelian submonoids of abelian-by-finite groups that aremaximal orders, starting from abelian maximal orders.

Proposition 4.2.1. Let A be an abelian normal subgroup of finite indexin a group G. Suppose that B is a submonoid of A so that A = BB−1 andB is a finitely generated maximal order. Let S be a submonoid of G suchthat G = SS−1 and S ∩ A = B. Then S is a maximal order that satisfiesthe ascending chain condition on right ideals if and only if S is maximalamong all submonoids T of G with T ∩A = B.

Proof. First suppose S is a maximal order that satisfies the ascending chaincondition on right ideals. Suppose that S ⊆ T ⊆ G for a submonoidT of G such that S ∩ A = B = T ∩ A. By assumption, B is finitelygenerated. Since also A is normal and of finite index in G, it thus followsfrom Lemma 3.1.5 that K[T ] is Noetherian and it is a finitely generatedright K[B]-module. Thus T satisfies the ascending chain condition on one-sided ideals and T =

⋃ni=1 tiB for some n ≥ 1 and ti ∈ T . Since G is

finitely generated and abelian-by-finite, we know from Lemma 3.2.1, thatfor every i there exists zi ∈ Z(S) such that ziti ∈ S. Let z = z1 · · · zk. Thenzti ∈ S and therefore zT =

⋃i zti(S ∩A) ⊆ S. Since S is a maximal order,

this implies that T = S. So, we have shown that S is maximal among allsubmonoids T ⊆ SS−1 such that T ∩A = S ∩A.

Conversely, assume that S ⊆ G is maximal among all submonoids Tof G with T ∩ A = B. As above, because S ∩ A is finitely generated, Ssatisfies the ascending chain condition on right ideals. Suppose that T ⊆ Gis a submonoid such that S ⊆ T and gTh ⊆ S for some g, h ∈ G. Thereexist s, t ∈ S such that sg, ht ∈ Z(G) ∩ B. So sgTht ⊆ S and thereforeTz ⊆ S for some z ∈ Z(G)∩B. In particular (T ∩A)z ⊆ S ∩A = B. SinceS ∩A ⊆ T ∩A and S ∩A is a maximal order, it follows that T ∩A = S ∩A.Because S ⊆ T , the assumption on S then implies that T = S. ThereforeS is a maximal order.

4.3 An illustration of the main theorem

In this section we give the promised illustration of Theorem 4.1.3. Thisexample also shows that Theorem 3.2.2 cannot be extended to prime idealsof height exceeding one.

The proof of this example makes use of Proposition 4.2.1. So we needto start with the following commutative example that contains the abelianmonoid generated by u1, u2, u3, u4 subject to the unique relation u1u2 =



u3u4. It was shown by Anderson in [2] that the latter is a cancellativemonoid that is a maximal order. We will often refer to this monoid as theAnderson monoid. In Chapter 5, we classify all finitely presented abelianmonoids defined by at most two relations that yield a cancellative maximalorder. However, to illustrate that proof, we give a proof for the followingexample.

Example 4.3.1. The abelian monoid B = 〈u1, u2, u3, u4, u5, u6〉 definedby the relations u1u2 = u3u4 = u5u6 is a cancellative monoid that is amaximal order (in its torsion free group of quotients).

Proof. Let F = 〈x1, . . . , x8〉 be a free abelian monoid of rank 8. Define

b1 = x1x2x3x4, b2 = x5x6x7x8, b3 = x1x2x5x6,

b4 = x3x4x7x8, b5 = x1x3x5x7, b6 = x2x4x6x8.

Clearly, these bi satisfy the defining relations for B. We claim that actuallyB ∼= 〈b1, . . . , b6〉 under the map determined by ui 7→ bi, i = 1, . . . , 6. Inorder to prove this, suppose that there is a relation w = v, where w, v arenontrivial words in bi. We need to show that this relation follows fromb1b2 = b3b4 = b5b6. Cancelling in F , if needed, we may assume that eachbi appears at most on one side of the relation. We also may assume thatnot both sides are divisible in 〈b1, . . . , b6〉 by one of the equal words b1b2,b3b4 and b5b6. Further, on both sides of v = w we need some bi with aneven i. Indeed, suppose the contrary, then x8 is not involved in v and w.Hence also b5 cannot occur because of x7. But, as b1 and b3 generate a freeabelian monoid of rank 2, it then follows that v and w are identical wordsin the bi’s, as desired. Hence, by symmetry we may assume that w containsbi22 with i2 > 0 and w does not contain b4 nor b6 as a factor, and v does notcontain b2 as a factor and contains bi44 b

i66 for some nonnegative i4, i6 with

i2 = i4 + i6 (the latter follows by taking into the account the degree of x8

in the respective words). Looking at x4 we then get that b1 appears in w.If i6 = 0 then (looking at x6) we get that b3 is in v, a contradiction becauseb1b2 = b3b4 divides then both v and w. So i6 > 0. Also v must containx1, x5 and so b3 or b5 is in v. The latter is not possible because then v andw are divisible by b1b2 = b5b6, a contradiction. Thus, b3 occurs in v. Thenwe must have i4 = 0 because otherwise b1b2 = b3b4 divides v and w. Sow = bi11 b

i22 b

i55 = bi33 b

i66 = v for some i1, i2, i6 > 0 and i3, i5 ≥ 0. Then the

exponents of x7 show that i2 = i5 = 0, a contradiction. The claim follows.Hence we may indeed identify ui with bi and B with 〈b1, . . . , b6〉.



Next, suppose that w = ui22 u

i33 u

i44 u

i55 ∈ F ∩ gr(B). This is equivalent

to the conditions: i3 + i5 ≥ 0, i3 ≥ 0, i4 + i5 ≥ 0, i4 ≥ 0, i2 + i3 + i5 ≥0, i2 + i3 ≥ 0, i2 + i4 + i5 ≥ 0, i2 + i4 ≥ 0. Let j = min{i3, i4} ≥ 0.Then j + i2 + i5 ≥ 0. We choose s, t ≥ 0 so that i2 + s, i5 + t ≥ 0 ands+ t = j. Then w = us

1ui2+s2 ui3−j

3 ui4−j4 ui5+t

5 ut6. It is thus clear that w ∈ B.

So B = F ∩ gr(B). Since F is a maximal order it then easily follows thatB is a maximal order.

We now give the answer to Example 3.1.1.


x1x4 = x2x3, x1x3 = x2x4, x3x1 = x4x2

x3x2 = x4x1, x1x2 = x3x4, x2x1 = x4x3.

Clearly, R = K[S] for the monoid S defined by the same presentation.Then S is cancellative (but the group SS−1 is not torsion free) and R is aprime Noetherian PI-algebra that is a maximal order. Furthermore, thereexists a prime ideal P of S so that K[P ] is not a prime ideal of K[S].

Proof. Notice that each of the permutations (12)(34), (13)(24) and (14)(23)determines an automorphism of S. First we list some equalities in S, namelyall relations between the elements of length 3. For brevity, we use the indexi in place of the generator xi.

112 = 134 = 244, 113 = 124 = 344, 114 = 123 = 343 = 321 = 411221 = 243 = 133, 224 = 213 = 433, 223 = 214 = 434 = 412 = 322331 = 342 = 122, 334 = 312 = 422, 332 = 341 = 121 = 143 = 233442 = 431 = 211, 443 = 421 = 311, 441 = 432 = 212 = 234 = 144

131 = 142 = 232 = 241, 141 = 132 = 242 = 231313 = 423 = 414 = 324, 323 = 314 = 424 = 413.

It follows easily that A = 〈x21, x

22, x

23, x

24〉 is an abelian submonoid and it is

normal, that is xA = Ax for every x ∈ S. Moreover x21x

24 = x2

2x23 because

x2x2x3x3 = x2x3x3x2 = x1x4x4x1 = x4x4x1x1.Let u1 = x1x4, u2 = x4x1, u3 = x2

1, u4 = x24, u5 = x2

2, u6 = x23 and

B = 〈u1, u2, u3, u4, u5, u6〉. From the above equalities it follows that B isabelian and sB = Bs for every s ∈ S.



Every element of S that is a word of length 3 is either of the form xyyor of the form xx1x4, xx4x1 for some x. It is also easy to see that everyelement of S is of the form zu1w1 · · ·urwr or zu1w1 · · ·urwrur+1 or zc,where z ∈ A and c ∈ S is an element of length at most 2 in the xj and eitherui ∈ {x1, x2}, wi ∈ {x3, x4} or ui ∈ {x3, x4}, wi ∈ {x1, x2} (for all i). So wemay assume that c ∈ {x1x2, x2x1}. Here r is a non-negative integer. Usingthe relations listed above (especially x1x4x1 = x2x4x2, x4x1x4 = x3x1x3) itmay be checked that the even powers of x1x4 = x2x3 and x2x4 = x1x3 areequal and the even powers of x4x1 = x3x2 and x3x1 = x4x2 are equal. Also{x4x1, x1x4}{x1x2, x2x1} ⊆ A{x1x3, x3x1}. It follows that possible formsof elements of S are

z(x1x4)i, z(x1x4)ix1, z(x1x4)ix2, z(x1x4)ix1x3, zx1x2 (4.1)z(x4x1)i, z(x4x1)ix4, z(x4x1)ix3, z(x4x1)ix3x1, zx2x1, (4.2)

with z ∈ A, i ≥ 0. This leads to

S = B ∪Bx1 ∪Bx2 ∪Bx3 ∪Bx4 ∪Bx1x3 ∪Bx3x1 ∪Bx1x2 ∪Bx2x1.(4.3)

It follows thatK[S] is finite module over the finitely generated commutativealgebra K[B]. Hence K[S] is a Noetherian PI-algebra.

Let C be the free abelian group of rank 4 generated by elements a, b, c, d.Let X be the free monoid on x1, x2, x3, x4. Consider the monoid homomor-phism φ : X −→M4(K[C]) defined by

x1 7→

0 a 0 01 0 0 00 0 0 a−1bc0 0 1 0

, x2 7→

0 0 b 00 0 0 a−1bc1 0 0 00 ab−1 0 0

,

x3 7→

0 0 bcd−1 00 0 0 a−1bd

b−1d 0 0 00 ad−1 0 0

, x4 7→

0 bcd−1 0 0

a−1d 0 0 00 0 0 d0 0 ad−1 0

.

It is easy to check that these matrices satisfy the defining relations of S,so φ can be viewed as a homomorphism from S to the group of monomial



matrices over C. Moreover

u3 7→

a 0 0 00 a 0 00 0 a−1bc 00 0 0 a−1bc

, u5 7→

b 0 0 00 c 0 00 0 b 00 0 0 c

u4 7→

a−1bc 0 0 0

0 a−1bc 0 00 0 a 00 0 0 a

, u6 7→

c 0 0 00 b 0 00 0 c 00 0 0 b

,

u1 7→

d 0 0 00 bcd−1 0 00 0 bcd−1 00 0 0 d

, u2 7→

bcd−1 0 0 0

0 d 0 00 0 d 00 0 0 bcd−1

.

The projection of the group C ′ generated by φ(u1), φ(u3), φ(u5), φ(u6) ontothe (1, 1)-entry contains a, b, c, d, whence it is free abelian of rank 4. SoC ′ ∼= C. In particular, φ is injective on B0 = 〈u1, u3, u5, u6〉, and thus B0

is a free abelian monoid of rank 4. Let B′ be the abelian monoid withpresentation 〈y1, y2, y3, y4, y5, y6 | y1y2 = y3y4 = y5y6〉. Clearly, we havenatural homomorphisms B′ −→ B −→ φ(B). We also know that φ(B)generates a free abelian group of rank ≥ 4 and B′ has a group of quotientsthat is free abelian of rank 4. So B′ and φ(B) must be isomorphic, sinceotherwise under the map B′ −→ φ(B) we have to factor out an additionalrelation and the rank would decrease. It follows that φ is injective on Band thus, because of Example 4.3.1, B is a cancellative maximal order withgroup of quotients N = BB−1 ∼= Z4. Note that AA−1 = gr(u3, u5, u6).Using the defining relations u1u2 = u3u4 = u5u6 for B, it is readily verifiedthat if i, j, k ∈ Z then ui

3uj5u

k6 ∈ A = 〈u3, u4, u5, u6〉 if and only if j, k ≥ 0

and min(j, k) + i ≥ 0.The images under φ of the first 4 types listed in (4.1) have different

patterns of nonzero entries in M4(K[C]). The same applies to the first 4types in (4.2). Notice that φ is injective on each of the 10 types. Sup-pose that s = z(x1x4)ix1, t = z′(x4x1)jx4 have the same image. Thensx4 = z(x1x4)i+1 and tx4 = z′(x4x1)jx2

4 have equal images. This is notpossible because (x1x4)i+1A ∩ (x4x1)jA = ∅ by the above description ofthe group BB−1 ∼= φ(B)φ(B)−1. Similarly one deals with elements of anytwo different types listed in (4.1) and (4.2), showing that only elementsthe form s = zx1x2, t = z′x2x1, where z, z′ ∈ A, can have equal images.



Then φ(zx1x1x4x1) = φ(tx4x2) = φ(sx4x2) = φ(z′x4x1x2x2) and can-cellativity of φ(B) ∼= B yields zx1x1 = z′x2x2. Write z = ui

3uj5u

k6, with

i, j, k ∈ Z. Since z, z′ ∈ A, the above yields that z′ = ui+13 uj−1

5 uk6 and

j, k ≥ 0,−i and j − 1, k ≥ 0,−(i + 1). Notice that u3x2x1 = x1x1x2x1 =x2x2x1x2 = u5x1x2. Therefore, if j− 1 ≥ −i then ui

3uj−15 uk

6 ∈ A and hencezx1x2 = ui

3uj−15 uk

6u5x1x2 = ui3u

j−15 uk

6u3x2x1 = z′x2x1. On the other hand,if j = −i then uj−1

4 uk−j6 ∈ A and u4x1x2 = u6x2x1. Hence we also get

zx1x2 = uj4u

k−j6 x1x2 = uj−1

4 uk−j6 u4x1x2 = uj−1

4 uk−j6 u6x2x1 = z′x2x1.

It follows that φ is injective on all elements of types (4.1),(4.2). There-fore φ is an embedding and thus S is cancellative.

We identify S with φ(S). Put G = SS−1. Then G = N ∪ x1N ∪ x2N ∪x1x2N . Moreover N ∼= Z4 and N is a normal subgroup with G/N the fourgroup. From (4.3) it follows that S ∩N = B.

We now show that G is dihedral free. To prove this, notice that S actsby conjugation on B and the generators xi of S correspond to the follow-ing permutations σi of the generating set of B (the numbers 1, 2, 3, 4, 5, 6correspond to the generators u1, u2, u3, u4, u5, u6).

σ1 = (12)(56), σ2 = (12)(34), σ3 = (12)(34), σ4 = (12)(56). (4.4)

Suppose D ⊆ G is an infinite dihedral group such that the normalizerNG(D) of D in G is of finite index. Let t ∈ D be an element of order 2.Then there exists k ≥ 1 such that uk

i tu−ki t ∈ D for i = 1, 2, . . . , 6. Clearly

uki tu

−ki t = uk

i ukσ(i)t

2 = uki u

kσ(i),

where σ is the automorphism of N determined by t. Since t ∈ Nx1 ∪Nx2 ∪Nx1x2, it follows that σ is determined by the conjugation by x1, x2

or x1x2. So σ permutes exactly two of the pairs u1, u2;u3, u4 and u5, u6.It follows that uk

i ukj , u

kpu

kq ∈ D for two different pairs i, j and p, q. Hence

rk(N∩D) ≥ 2, a contradiction. Therefore G indeed is a dihedral free group.Next we show that ∆+(G) is trivial. For this, let F be a finite normal

subgroup of G. Since N is torsion free, it is clear that F is isomorphic witha subgroup of Z2 × Z2. A nontrivial element t ∈ F must be of order 2,whence as above we get that un

i unj t ∈ F for some i 6= j and every n ≥ 1.

Therefore F is infinite, a contradiction. It follows that ∆+(G) is trivial.Therefore K[G] is prime and hence K[S] also is prime.

Note that in G we have x−12 x1 = x3x

−14 = x4x

−13 and x1x

−12 = x−1

3 x4 =x−1

4 x3. So x−12 x1 is an element of order 2.



We describe the minimal prime ideals of S. For this we first notice thatit is easy to see that the minimal prime ideals of B are:

Q = Q1 = (u1, u3, u5), Q2 = Qx1 = (u2, u3, u6),Q3 = Qx2 = (u2, u4, u5), Q4 = Qx1x2 = (u1, u4, u6),Q′ = Q5 = (u2, u3, u5), Q6 = (Q′)x1 = (u1, u3, u6),

Q7 = (Q′)x2 = (u1, u4, u5), Q8 = (Q′)x1x2 = (u2, u4, u6).

Because of (4.4), it is easily verified that u1u2 is a central element of S andthat every ideal of S contains a positive power of u1u2. In particular, u1u2

belongs to every prime ideal of S. Consider in B the following G-invariantideals:

M = (u1u3u5, u2u3u6, u2u4u5, u1u4u6, u1u2)

andM ′ = (u2u3u5, u1u4u5, u1u3u6, u2u4u6, u1u2).

Again because of (4.4), it is easy to see that bSb′ ⊆ u1u2S for every defininggenerator b of M and b′ of M ′. It follows that a prime ideal of S containsM or M ′.

Notice that xu1u3u5y ∈ xy{u1u3u5, u2u4u5, u2u3u6, u1u4u6} for everyx, y ∈ S. Therefore Su1u3u5S ∩ 〈u2, u3, u5〉 = ∅ (for example, xyu1u3u5 6∈〈u2, u3, u5〉 because otherwise xy = u−1

1 u−13 u−1

5 〈u2, u3, u5〉 ∩ (S ∩N), whichis not possible because S∩N = B and N = gr(u1, u2, u3, u5) is free abelianof rank 4). So there exists a (unique) ideal P of S that is maximal withrespect to the property P ∩ 〈u2, u3, u5〉 = ∅. It is easy to see that P is aprime ideal of S. Since u2u3u5 6∈ P , we get that M ′ 6⊆ P , whence M ⊆ P .

Let E = u2u3u5〈u2, u3, u5〉. Then

x4〈u1, u3, u6〉x1 ⊆ E, x2〈u1, u4, u5〉x2 ⊆ E, x1x3〈u2, u4, u6〉x3x1 ⊆ E.(4.5)

Therefore P ∩ 〈u1, u3, u6〉 = ∅, P ∩ 〈u1, u4, u5〉 = ∅, P ∩ 〈u2, u4, u6〉 = ∅. Forevery element b ∈ B \M the ideal bB intersects one of the sets

u2u3u5〈u2, u3, u5〉, u1u3u6〈u1, u3, u6〉, u1u4u5〈u1, u4, u5〉, u2u4u6〈u2, u4, u6〉.

Since these sets do not intersect P , we get that P ∩B = M . Hence it is G-invariant and, becauseM ⊆ Q1∩Q2∩Q3∩Q4 and all Qi are minimal primesof B, Corollary 3.2.5 yields that P∩B = M = Q1∩Q2∩Q3∩Q4 = B(SMS)and P is a minimal prime ideal of S. A similar argument shows thatthere exists an ideal P ′ of S that is maximal with respect to the property



P ′∩〈u1, u3, u5〉 = ∅, and it follows that P ′∩B = M ′ = Q5∩Q6∩Q7∩Q8 =B(SM ′S) is the only other minimal prime of S.

In order to continue the proof we first show the following claim on therepresentation of elements of B.

Claim: PresentationAn element b = uα1

1 uα33 uα4

4 uα66 of N = BB−1 is in B (with each αi ∈ Z) if

and only if α3, α4 ≥ 0 and either (i) α1 ≥ 0 and min(α3, α4)+α6 ≥ 0 or (ii)α1 < 0, α3 +α1, α4 +α1 ≥ 0 and min(α3, α4)+α1 +α6 ≥ 0, or equivalently,min(α3 + α1, α4 + α1) ≥ max(0,−α6).

That the first condition is sufficient is easily verified. For the secondcondition, we rewrite uα1

1 as uα13 uα1

4 u−α12 and the result follows because of

the first part (by interchanging u1 with u2).To prove that they are necessary, suppose b = uα1

1 uα33 uα4

4 uα66 ∈ B, with

each αi ∈ Z. Because Bu4 ∩ 〈u1, u3, u6〉 = ∅ and Bu3 ∩ 〈u1, u4, u6〉 = ∅, itfollows that α3, α4 ≥ 0. Suppose now that α1 ≥ 0. Then

b = uα11 u

α3−min(α3,α4)3 u


min(α3,α4)5 u

α6+min(α3,α4)6

and thus uα11 u



min(α3,α4)5 ∈ Bu−(α6+min(α3,α4))

6 .Since the exponent of u3 or u4 is 0, this implies that α6 + min(α3, α4) ≥ 0,as desired.

On the other hand, suppose that α1 < 0. Then b = u−α12 uα3+α1

3 uα4+α14 uα6

6 .The previous case yields that α3+α1 ≥ 0, α4+α1 ≥ 0 and min(α3+α1, α4+α1) + α6 ≥ 0, again as desired. This proves the claim Presentation.

Next we show that S is a maximal order. First note that B is a maximalorder by Example 4.3.1. So, because of Proposition 4.2.1, it is sufficient toprove that if s ∈ SS−1 \ S then 〈S, s〉 ∩N strictly contains B.

We know that G = SS−1 = N ∪x1N ∪x2N ∪x1x2N . If s ∈ N \B thenclearly 〈S, s〉 ∩N strictly contains B. So, there are three cases to be dealtwith: (1) s = bx1 ∈ G \ S, (2) s = bx2 ∈ G \ S, and (3) s = bx1x2 ∈ G \ S,where b ∈ N \B.

Case (1): s = bx1 ∈ G \ S. Obviously, bx1x4 = bu1 ∈ 〈S, s〉. Ifbu1 ∈ N \ B, then we are done. So suppose bu1 ∈ B and b ∈ N \ B. Wewrite b in the following form:

uα11 uα3

3 uα44 uα6

6 .

First, suppose that α1 ≥ 0. By the claim Presentation, from bu1 ∈ B weget that min(α3, α4) ≥ max(−α6, 0). But, also from b ∈ N \ B we getthat min(α3, α4) < max(−α6, 0), a contradiction. So, α1 has to be strictlynegative.



Therefore, assume that α1 < 0. Then

b = uα11 uα3

3 uα44 uα6

6 = u−α12 uα3+α1

3 uα3+α14 uα6

6

andbu1 = u−α1−1

2 uα3+α1+13 uα3+α1+1

4 uα66 .

By the claim Presentation (by interchanging u1 and u2), we get that min(α3+α1 + 1, α4 + α1 + 1) ≥ max(−α6, 0), but also min(α3 + α1, α4 + α1) <max(−α6, 0). Hence, min(α3 + α1 + 1, α4 + α1 + 1) = max(−α6, 0). Sup-pose α6 ≥ 0. Then min(α3 + α1 + 1, α4 + α1 + 1) = 0. Therefore,

b = uα11 uα3

3 uα44 uα6

6 = u−11 uα3

2 uα4−α34 uα6

6

and α4 > α3 orb = u−1

1 uα42 uα3−α4

3 uα66

and α3 ≥ α4. In the first case we get that bx1 ∈ S, since x−14 x−1

1 x4x4x1 =x4. So this case gives a contradiction and is hence impossible. In the secondcase, bx1x1 = u−1

1 uα42 uα3−α4+1

3 uα66 ∈ (〈S, s〉 ∩N) \B, as desired. If α6 < 0,

min(α3 + α1 + 1, α4 + α1 + 1) = −α6 and therefore

b = uα11 uα3

3 uα44 uα6

6 = u−11 u−α1−1

2 uα3−α43 u−α6

5

if α4 ≤ α3 orb = u−1

1 u−α1−12 uα4−α3

4 u−α65

if α3 < α4. Since −α1 − 1 ≥ 0 and −α6 > 0, by interchanging u5 and u6,this case is completely similar to the case where α6 ≥ 0. This finishes theproof of Case (1).

Case (2): s = bx2 ∈ G \ S. The permutation σ = (12)(34) determinesan automorphism σ on S with σ(B) = B. Clearly, σ(s) = b′x1 withb′ = σ(b) 6∈ B. From Case (1) we get that 〈S, σ(s)〉 ∩N properly containsB. Again applying σ to the latter we get that 〈S, s〉 ∩N properly containsB, as required.

Case (3): s = bx1x2 ∈ G \ S. Clearly, bx1x2x2x4 = bu1u6 ∈ 〈S, s〉. Ifbu1u6 ∈ N \ B, then we are done. So suppose bu1u6 ∈ B and b ∈ N \ B.Write

b = uα11 uα3

3 uα44 uα6

6 .

We know that α3, α4 ≥ 0 and we consider again the two cases: α1 ≥ 0 andα1 < 0, separately.



First assume that α1 ≥ 0. Since bu1u6 ∈ B and b 6∈ B, we get thatmin(α3, α4) = −α6 − 1. Then

b = uα11 uα3

3 uα44 uα6

6 = uα11 uα3−α4

3 u−α6−15 u−1

6 ,

if α4 ≤ α3, orb = uα1

1 uα4−α34 u−α6−1

5 u−16 ,

if α3 < α4. Now, let α1 > 0. Then bx1x2 ∈ S, since x1x4x−13 x−1

3 x1x2 =x2x4, a contradiction. So this case is impossible. Therefore, α1 = 0. Ifα3 < α4, then bx1x2 ∈ S, since x4x4x

−13 x−1

3 x3x4 = x2x1, a contradiction,so this case is again impossible. If α4 ≤ α3, then bx1x2x3x1 = bu2u3 ∈(〈S, s〉 ∩N) \B, as desired.

Finally, suppose α1 < 0. Then

b = uα11 uα3

3 uα44 uα6

6 = u−α12 uα3+α1

3 uα4+α14 uα6

6

and bu1u6 = u−α1−12 uα3+α1+1

3 uα4+α1+14 uα6+1

6 . By the claim Presentation(by interchanging u1 and u2), we get that min(α3 + α1 + 1, α4 + α1 + 1) ≥max(−α6− 1, 0), but also min(α3 +α1, α4 +α1) < max(−α6, 0). If α6 ≥ 0,min(α3, α4) + α1 = −1 and

b = u−11 uα4

2 uα3−α43 uα6

6 ,

if α4 ≤ α3, orb = u−1

1 uα32 uα4−α3

4 uα66 ,

if α3 < α4. So, we have the same conditions here as in Case 1 and the resultfollows. If α6 < 0, there are two possibilities: min(α3, α4) + α1 = −α6 − 1or min(α3, α4) + α1 = −α6 − 2. In the first case, we get that

b = u−11 u−α1−1

2 uα3−α43 u−α6

5

if α4 ≤ α3, orb = u−1

1 u−α1−12 uα4−α3

4 u−α65

if α3 < α4. Since −α1 − 1 ≥ 0 and −α6 > 0, by interchanging u5 andu6, this case is completely similar to the case where α6 ≥ 0 and hence canalso be treated as in Case 1. Finally, if the second possibility holds, that ismin(α3, α4) + α1 = −α6 − 2, then

b = u−21 u−2−α1

2 uα3−α43 u−α6

5 ,

if α4 ≤ α3, orb = u−2

1 u−2−α12 uα4−α3

4 u−α65 ,



if α4 > α3, so we always get that bu2u3 ∈ (〈S, s〉 ∩N) \B, as desired.This finishes the proof of the fact that S is a maximal order. Since

P ∩ B is invariant for every minimal prime P of S, it then follows fromTheorem 4.1.3 that K[S] is a maximal order.

Let V be the ideal of S generated by the elements u3, u4, u5, u6, x1x2, x2x1.It easily follows that the elements of S \ V are of the form

(x1x4)i, (x1x4)ix1, (x1x4)ix2, (x1x4)ix1x3,

(x4x1)i, (x4x1)ix4, (x4x1)ix3, (x4x1)ix3x1,

where i is a non-negative integer. Then S \ V = {1} ∪ I where the set Ican be written in matrix format as a union of disjoint sets:(

I11 I12I21 I22

),

with I11 = 〈u2〉u2 ∪ 〈u2〉x3x1, I12 = 〈u2〉x3 ∪ 〈u2〉x4, I21 = x1〈u2〉 ∪ x2〈u2〉and I22 = 〈u1〉u1∪〈u1〉x1x3. Moreover, IijIkl ⊆ Iil if j = k, and is containedin V otherwise. In S/V the set I ∪ {0} is an ideal and the semigroup I11(treated as a subsemigroup of S) has a group of quotients H = gr(u2, x3x1).Since u2

2 = (x3x1)2 and u2(x3x1) = (x3x1)u2, we get that H is isomorphicwith Z× Z2. From the matrix pattern of I it follows that S/V is a primesemigroup. So V is a prime ideal of S. However, because K[H] and thusK[I11] is not prime, standard generalized matrix ring arguments yield thatK[S]/K[V ] is not prime.


Chapter 5

Abelian maximal orders andclass groups

In Chapters 1 and 3 we already mentioned that Anderson [1, 2], for abelianmonoids S, characterized when K[S] is a prime Noetherian maximal order.More generally, Chouinard [14] described commutative semigroup algebrasK[S] that are Krull domains. Indeed, in [1, 2, 14, 28], the authors provedthat an abelian Noetherian semigroup algebra K[S] is an integrally closeddomain if and only if the finitely generated abelian monoid S is integrallyclosed in its torsion free group of quotients G = SS−1. So, integral closed-ness of K[S] is a homogeneous property, i.e., a condition on the monoid S.Furthermore, it was shown that cl(K[S]), the class group of K[S], is natu-rally isomorphic with cl(S), the class group of S, and thus all height oneprimes of K[S] are principal if and only all minimal primes of S are princi-pal. As an application, one obtained much easier calculations for the classgroup of several classical examples of Noetherian integrally closed domains.For non-commutative monoids S, this problem is unsolved. For example,for binomial semigroups (see [39, 45]) one knows that cl(S) = {1}, but it isunknown whether cl(K[S]) = {1}.

Because of our previous chapters (in particular Proposition 4.2.1), it isrelevant to determine when a finitely generated cancellative abelian monoidS is a maximal order. Recall that Chouinard in [14] obtained such a de-scription.

Theorem 5.0.1. A finitely generated cancellative abelian monoid S isa maximal order in its group of quotients SS−1 if and only if

S ∼= U(S)× S1,

67

5.1. Finitely presented maximal orders with one relation 68

where U(S) is the group of invertible elements of S and S1 is a submonoidof a free abelian group F so that S1 = S1S

−11 ∩F+, with F+ a positive cone

of F. Furthemore, if SS−1 is torsion free, then S is a maximal order if andonly if K[S] is an integrally closed domain.

Despite this nice and useful structural characterization, the followingremains a challenging problem for a finitely generated submonoid S of anabelian group: determine necessary and sufficient conditions on the definingrelations for S to be a maximal order.

In this chapter we precisely solve the problem in case S is defined byone or two relations (Theorem 5.1.2 and Theorem 5.2.2). So, in contrast tothe structural description mentioned before, our contribution is a computa-tional approach (based on presentations) to obtain a description of maximalorders defined via monomial relations. The advantage is that it also allowsus to compute the class group cl(S) of such monoids S and thus of theiralgebras K[S] (Theorem 5.1.4 and Theorem 5.2.4). This group is the basictool in the study of arithmetics of maximal orders ([23]) and hence theseresults might be interesting for experts in related areas. In Section 5.3 weconclude with some comments and examples on monoids defined by morethan two relations. These indicate that the results cannot be extended.The results in this chapter will appear in [32].

5.1 Finitely presented maximal orders with onerelation

Let S be a monoid within a group of quotients G that satisfies the ascendingchain condition on two-sided ideals. Recall from Lemma 2.5.10, that anorder S is a maximal order in G if and only if S = (P :l P ) = (P :r P ) forevery prime ideal P of S.

For an abelian cancellative monoid S we will often use another charac-terization via minimal prime ideals Q. By SQ we denote the localizationof S (within its group of quotients G) with respect to the multiplicativelyclosed set S \ Q. It is well known that if S = 〈s1, . . . , sn〉, so S is finitelygenerated, then it is a maximal order if and only if S =

⋂Q SQ, where

the intersection runs over all minimal prime ideals of S, and each SQ isa discrete valuation semigroup. Furthermore, if SQ is a discrete valuationsemigroup then SQ = U(SQ)〈si〉 for some 1 ≤ i ≤ n.

Our main aim is to describe when a finitely generated abelian monoidwhich is defined by at most two relations is a maximal order. A first

Abelian maximal orders and class groups


important obstacle to overcome is to determine when such monoids arecancellative, i.e., when they are contained in a group. Because of the com-ments given in the introduction, and since we are mainly interested in suchmonoids that are maximal orders, we only need to deal with monoids Sso that U(S) = {1}. In this context we mention that, in [11] and [24], analgorithm of Contejean and Devie is used to determine whether a finitelygenerated monoid given by a presentation is cancellative.

We will use the following notation. By FaMn we denote a free abelianmonoid of rank n. If FaMn = 〈u1, . . . , un〉 and w = ua1

1 · · ·uann ∈ FaMn,

then put supp(w) = {ui | ai 6= 0}, the support of w. Now, suppose S has apresentation

S = 〈u1, . . . , un | w1 = v1, . . . , wm = vm〉,

where wi, vi are words in the free abelian monoid FaMn = 〈u1, . . . , un〉.Clearly, if S is cancellative, then we may assume it has a presentation with

supp(wi) ∩ supp(vi) = ∅,

for all i.

Proposition 5.1.1. Let S be an abelian monoid defined by the presentation

〈u1, . . . , un | u1 · · ·uk = uak+1

k+1 · · ·uann 〉

for some positive integers ak+1, . . . , an and some k < n. Let FaMk(n−k) =〈xi,j | 1 ≤ i ≤ k, 1 ≤ j ≤ n − k〉, a free abelian monoid of rank k(n − k).For 1 ≤ j ≤ k put

vj = xak+1

j,1 xak+2

j,2 · · ·xanj,n−k,

and for k + 1 ≤ j ≤ n put

vj = x1,j−kx2,j−k · · ·xk,j−k.

Then S ∼= V = 〈v1, . . . , vn〉 ⊆ FaMk(n−k) and 〈v1, . . . , vn〉 has v1 · · · vk =v

ak+1

k+1 · · · vann as its only defining relation.

Proof. Let V = 〈v1, . . . , vn〉 ⊆ FaMk(n−k). Clearly, v1 · · · vk = vak+1

k+1 · · · vann

and thus V = 〈v1, . . . , vn〉 is a natural homomorphic image of S.Since all ai 6= 0, it is easy to see that every relation in V (with disjoint

supports with respect to the vi’s) must involve all generators vi. More-over, since v1, vk+1 are the only generators involving x1,1, it follows thatin such a relation v1, vk+1 are on opposite sides of the equality. And also



vk+2, . . . , vn must be on the side opposite to v1 (look at the appearanceof x1,2, x1,3, . . . , x1,n−k in order to see this). Similarly, by looking at theappearance of x21, x31, . . . , xk1, we get that v2, . . . , vk must be on the sideopposite to vk+1. It follows that every relation in V , possibly after cancel-lation, must be of the form

vc11 · · · vck

k = vck+1

k+1 · · · vcnn (5.1)

for some positive integers cj . Again, using the fact that xi,j ’s are inde-pendent and comparing the exponent of xi,j on both sides of (5.1), weget that ak+jci = ck+j for 1 ≤ i ≤ k and j = 1, 2, . . . , n − k. Thisimplies that c1 = c2 = · · · = ck. Hence relation (5.1) is of the form(v1 · · · vk)c1 = (vak+1

k+1 · · · vann )c1 . So it is a consequence of the relation defin-

ing S with every uj replaced by vj . It follows that V ∼= S.

We now describe the defining relation of a one-relator finitely generatedmonoid S that is a maximal order (with U(S) = {1}). We also give arepresentation of such S as a positive cone of a subgroup of a free abeliangroup (see Theorem 5.0.1.). In order to do this, it is convenient to introducethe following notation. For a word w = ua1

1 · · ·uann in a free abelian monoid

FaMn = 〈u1, . . . , un〉 we put

Hsupp(w) = {uj | aj > 1}.

Theorem 5.1.2. Let S be the abelian monoid defined by the presentation

〈u1, . . . , un | w1 = w2〉,

with nonempty words w1 = ua11 · · ·uak

k , w2 = uak+1

k+1 · · ·uann , where k < n, and

each ai is a nonnegative integer. Then the semigroup S is a maximal orderin its torsion free group of quotients (or equivalently, by Theorem 5.0.1.,the semigroup algebra K[S] is an integrally closed domain) if and only ifHsupp(ua1

1 · · ·uakk ) = ∅ or Hsupp(uak+1

k+1 · · ·uann ) = ∅.

Proof. Let FaMn = 〈u1, . . . , un〉, a free abelian monoid of rank n. Wewrite vj for the image of uj ∈ FaMn in S. Then S = S′ ×FaMr, where thegenerators vj such that aj = 0 form a free basis of FaMr and S′ is generatedby the remaining vj ’s. Hence S′ is a maximal order if and only if S is amaximal order. Therefore we may assume that aj > 0 for every j.



First assume that S is a maximal order. Let aj = max{ai | 1 ≤ i ≤ n}.Because of symmetry, we may assume that j = k+ 1 and we need to provethat ai = 1 for 1 ≤ i ≤ k. Now,

((v1v−1k+1)v2 · · · vk)ak+1 = v

ak+1

1 v−ak+1

k+1 vak+1

2 · · · vak+1

k

= va11 v

a22 · · · vak

k vak+1−a1

1 · · · vak+1−ak

k v−ak+1

k+1

= vak+2

k+2 · · · vann v

ak+1−a1

1 · · · vak+1−ak

k ∈ S.

Since S is a maximal order, it follows that (v1v−1k+1)v2 · · · vk ∈ S and thus

v1v2 · · · vk ∈ vk+1S. Therefore, in the defining relation we need that a1 =a2 = · · · = ak = 1, as desired.

Conversely, assume that a1 = · · · = ak = 1. Hence, we may identify Swith the submonoid V of FaMk(n−k) described in Proposition 5.1.1. There-fore, it is sufficient to check (Theorem 5.0.1.) that V = V V −1∩FaMk(n−k).So, suppose v ∈ V V −1 ∩ FaMk(n−k). Then we may write

v = vc22 · · · vcn

n ∈ FaMk(n−k) (5.2)

for some integers ci. Looking at the exponent of x11 in (5.2) we get thatck+1 ≥ 0. Similarly, the exponents of the remaining x1j show that ci ≥ 0 forall i = k+1, . . . , n. Suppose that ci < 0 for some i. Let c = max{|ci| | ci <0}. Clearly, ci + c ≥ 0 for i = 2, . . . , k. Moreover, looking at the exponentof xj,m in (5.2) we get that ck+m + cjak+m ≥ 0 for every j = 1, . . . , k andevery m = 1, . . . , n − k. Hence ck+m − cak+m ≥ 0. This and the relationv1 · · · vk = v

ak+1

k+1 · · · vann imply that

v = vc1v

c2+c2 · · · vck+c

k vck+1−cak+1

k+1 · · · vcn−cann ∈ V.

So V V −1∩FaMk(n−k) ⊆ V . Hence, indeed, V = V V −1∩FaMk(n−k) follows.Clearly, the group SS−1 is free abelian of rank n − 1 and thus K[S] is anintegrally closed domain.

As an immediate application of Theorem 5.1.2, we get that the abelianAnderson monoid 〈u1, u2, u3, u4 | u1u2 = u3u4〉 is a maximal order. An-derson showed in [2] that the class group of this monoid is isomorphic toZ. This will follow also at once from Theorem 5.1.4, in which we describethe class group of any finitely generated one-relator abelian maximal order.For simplicity we assume that all generators are involved in the definingrelation. Because cl(FaMm) is trivial, this assumption is not restrictive. Wewill use the same notation for the generators ui of the free monoid FaMn

and for their images in S, if unambiguous.



Lemma 5.1.3. Let S = 〈u1, . . . , un | u1 · · ·uk = uak+1

k+1 · · ·uann 〉 be an abelian

maximal order and let Pyz denote the minimal prime ideal of S that isgenerated by the set {uy, uz}, where y ∈ {1, . . . , k}, z ∈ {k + 1, . . . , n}.Then

Suz = P1z ∗ · · · ∗ Pkz,

for every z ∈ {k + 1, . . . , n} and

Suy = Pak+1

yk+1 ∗ · · · ∗ Panyn ,

for every y ∈ {1, . . . , k}.

Proof. It is clear that the minimal primes of S are as described as inthe statement of the lemma. First, assume z ∈ {k + 1, . . . , n}. BecauseP1z, . . . , Pkz are k different minimal primes, it follows that P1z ∗ · · · ∗Pkz =P1z ∩ · · · ∩ Pkz. As the intersection consists of elements that are eitherproducts of generators with uz involved or the full product u1 · · ·uk =u

ak+1

k+1 · · ·uazz · · ·uan

n , it follows that P1z∩· · ·∩Pkz = (u1, uz)∩· · ·∩(uk, uz) =Suz.

Second, assume that y ∈ {1, . . . , k}. Note that

Pak+1

yk+1 ∗ · · · ∗ Panyn = (P ak+1

yk+1 · · ·Panyn )∗ = (S : (S : (P ak+1

yk+1 · · ·Panyn ))).

Because of uak+1

k+1 · · ·uann = u1 · · ·uy · · ·uk ∈ P

ak+1

yk+1 · · ·Panyn , it easily is veri-

fied thatP

ak+1

yk+1 · · ·Panyn ⊆ Suy,

and hence(S : (S : (P ak+1

yk+1 · · ·Panyn ))) ⊆ Suy.

On the other hand, since uak+1+···+any ∈ P

ak+1

yk+1 · · ·Panyn , it follows that

(S : (P ak+1

yk+1 · · ·Panyn )) ⊆ Su−l

y , for some positive integer l. We claim that onemay take l = 1. We show this by contradiction. So suppose there exists anelement g ∈ (S : (P ak+1

yk+1 · · ·Panyn )) ⊆ Su−l

y , such that g /∈ Su−1y . Hence g =

su−l′y , with s /∈ Suy and l′ ≥ 2. Because uak+1

k+1 · · ·uann = u1 · · ·uy · · ·uk ∈

Pak+1

yk+1 · · ·Panyn , we get that su−l′

y u1 · · ·uy · · ·uk = su−l′+1y t ∈ S, where t =

u1 · · ·uy−1uy+1 · · ·uk. Consequently, st ∈ Sul′−1y . Since s /∈ Suy, the defin-

ing relation implies that st /∈ Suy and hence l′−1 = 0, a contradiction. Thisproves the claim. Therefore we obtain that (S : (P ak+1

yk+1 · · ·Panyn )) ⊆ Su−1

y

and hence Suy ⊆ (S : (S : (P ak+1

yk+1 · · ·Panyn ))), which completes the proof.



Note that from the lemma it follows that, if k = 1, the minimal primesof S are principal. In particular, in this case, it follows that cl(S) = {1}.

We now describe the class group of a general one-relator monoid S.Let w = ui1 · · ·uik ∈ S. Then Sw = Sui1 ∗ · · · ∗ Suik in the group D(S).Every Suj in D(S) is a (unique) product of certain minimal primes Pj ,which leads to a unique presentation of Sw as a product of generators ofthe free group D(S). Therefore, every relation in cl(S) is a consequence ofthe relations obtained from the presentation of each Suj as a product ofsome Pl. This will be used in the following proof and also in the proof ofTheorem 5.2.4.

Theorem 5.1.4. Let S = 〈u1, . . . , un | u1 · · ·uk = uak+1

k+1 · · ·uann 〉 be an

abelian maximal order (with all ai > 0). Then

cl(K[S]) ∼= cl(S) ∼= Zk(n−k)−(n−1) × (Zd)k−1,

where d = gcd(ak+1, . . . , an), k(n− k) is the number of minimal primes inS and n− 1 is the torsion free rank of SS−1. In particular, if d = 1, thenthe class group of S is torsion free.

Proof. Clearly, because of the comment before the theorem, the result istrue for k = 1. So assume now that k ≥ 2. Recall that cl(S) ∼= D(S)/P (S)and that D(S) is the free abelian group with the basis consisting of theminimal primes of S. As there are k(n− k) minimal primes Pyz in S (with1 ≤ y ≤ k, k + 1 ≤ z ≤ n), we get that D(S) ∼= Zk(n−k). On the otherhand, P (S) = gr(Suy, Suz | y ∈ {1, . . . , k}, z ∈ {k + 1, . . . , n}). Hence, byLemma 5.1.3,

P (S) = gr(

(k∏

l=1

Plz

)∗,

(n∏

l=k+1

P alyl

)∗| y ∈ {1, . . . , k}, z ∈ {k + 1, . . . , n}).

In the proof we will abuse notation by simply writing(∏

P alyl

)∗as∏P al

yl .For z ∈ {k + 1, . . . , n} and y ∈ {1, . . . , k}, put

Qz =k∏

l=1

Plz and Qy =n∏

l=k+1

Pαlyl ,

where αl is such that αld = al. So,

cl(S) = gr(Pyz | y ∈ {1, . . . , k}, z ∈ {k + 1, . . . , n})/ gr(Qz, Q

dy | 1 ≤ y ≤ k, k + 1 ≤ z ≤ n).



For k + 1 ≤ z ≤ n, we have in cl(S)

P1z =

(k∏

l=2

Plz

)−1

, (5.3)

and thus, in particular,

Q1 =n∏

l=k+1

Pαl1l =

n∏l=k+1

k∏y=2

Pyl

−αl

=k∏

y=2

(n∏

l=k+1

P−αlyl

)=

k∏y=2

Q−1y (5.4)

Hence, because of (5.3), cl(S) is generated by Pyz with y ∈ {2, . . . , k} andz ∈ {k + 1, . . . , n}. Using (5.4), one then easily verifies that

cl(S) = gr(Pyz | y ∈ {2, . . . , k}, z ∈ {k + 1, . . . , n})/ gr(Qdy | 2 ≤ y ≤ k).

For y 6= y′ ∈ {2, . . . , k} we have that supp(Qy) ∩ supp(Qy′) = ∅. Hence, iteasily is seen that

cl(S) =k∏

y=2

gr(Pyz | z ∈ {k + 1, . . . , n})/ gr(Qdy).

Consider the subgroup Hy = gr(Qy) of the free abelian group Fy =gr(Pyz | z ∈ {k+ 1, . . . , n}) with basis {Pyz | z ∈ {k+ 1, . . . , n}}. We claimthat Fy/Hy is a torsion free group, and hence that cl(S) is torsion free,provided that d = 1. In order to prove this, we need to show that Fy/Hy

does not contain elements of order p, for any prime p.Suppose the quotient group has an element

n∏z=k+1

Pyzγz

of prime order p, with Pyz denoting the image of Pyz in Fy/Hy. In particularwe have that

∏nz=k+1 P

γzyz /∈ gr(Qy). Then(

n∏z=k+1

P γzyz

)p

=

(n∏

z=k+1

Pαzyz

)m

,

for some m ∈ Z. But this implies that, for every z ∈ {k + 1, . . . , n}:

γzp = αzm.


5.2. Finitely presented maximal orders with two relations 75

Since p is prime, either p | m or p | αz for every z. The latter is impossibleas gcd(αk+1, . . . , αn) = 1. Thus p | m. But then it follows that

n∏z=k+1

P γzyz ∈ gr(Qy),

a contradiction. So indeed, Fy/Hy is torsion free. Clearly we then havethat Fy/Hy is free abelian of rank n− k− 1. Finally, it easily follows fromthe defining relation that Qy yields an element of order d in Fy/ gr(Qd

y).Again, since Fy/Hy is torsion free, we get that

Fy/ gr(Qdy) ∼= Zn−k−1 × Zd.

Clearly, (n− k − 1)(k − 1) = k(n− k)− (n− 1) and therefore the proof isfinished.

5.2 Finitely presented maximal orders with tworelations

In this section we obtain a characterization of finitely presented monoidsthat are maximal orders and that are defined by two relations. The classgroup of such monoids S, and therefore of the corresponding algebras K[S],is also determined. We start with some consequences of cancellativity.

Lemma 5.2.1. Let S = 〈u1, . . . , un〉 be a finitely presented abelian monoiddefined by two independent relations

w1 = w2 and w3 = w4,

where all wi are nonempty words. If S is cancellative, then supp(w1) ∩supp(w2) = ∅, supp(w3) ∩ supp(w4) = ∅ and there exists at least one i ∈{1, 2, 3, 4} such that supp(wi) ∩ supp(wj) = ∅, for all j ∈ {1, 2, 3, 4} \ {i}.

Proof. To prove the first part of the statement, assume supp(w1)∩supp(w2)6= ∅. Hence w1 = xy and w2 = xz (as words) for some words x, y, z withsupp(x) 6= ∅ and supp(y) ∩ supp(z) = ∅. Moreover y or z is nonempty.Since S is cancellative, it follows that y = z in S. Hence, both wordsy and z are nonempty, since otherwise we get that the unit group U(S)is nontrivial which is not possible because all wi are nonempty. Henceneither of w1, w2 is a subword of y or of z. Thus, y = z in S implies thatw3 is a subword of y and w4 is a subword of z (or the symmetric case).



Let k be the maximal positive integer so that y = wk3 and z = wk

4z′ (as

words) for some y′ and z′. Then, in S, we get that y′ = z′ and eithery′ or z′ is not rewritable in S. Whence y′ and z′ are equal words. Sincesupp(y) ∩ supp(z) = ∅, we get that y = wk

3 and z = wk4 (as words). The

relation w1 = w2 thus looks like: xwk3 = xwk

4 . So, it is a consequenceof the second relation. This yields that the relations are dependent, acontradiction. Hence supp(w1) ∩ supp(w2) = ∅. As a similar argumentapplies also to the relation w3 = w4, this finishes the proof of the first partof the statement. As a consequence, we can write the defining relations asfollows

x1y1d1 = v2z2d2, x3v3d3 = y4z4d4, (5.5)

with xi ∈ X, yi ∈ Y , vi ∈ V , zi ∈ Z, di ∈ Di where X,Y, V, Z,D1, D2, D3

and D4 are submonoids of S that are generated by disjoint subsets of thegenerating set {u1, . . . , un} of S.

We now prove the second part of the statement. Suppose supp(x1) ∩supp(x3) 6= ∅. So, write

x1 = xx1 x3 = xx3, (5.6)

with x1, x3, x1, x3, x ∈ X, x 6= 1, and supp(x1) ∩ supp(x3) = ∅. We willprove that supp(z2)∩ supp(z4) = ∅. Assume, the contrary, that is, supposethat z2 = zz′2 and z4 = zz′4, where z, z′2, z

′4 ∈ Z, 1 6= z, supp(z′2)∩supp(z′4) =

∅ and supp(z) 6= ∅ (so z′4 is a proper subword of z4 and z′2 is a propersubword of z2). Clearly,

z2z′4 = z′2z4. (5.7)

The relations (5.5) and (5.7) imply the following equality in S

x1y1d1y4z′4d4 = v2z2d2y4z

′4d4 = v2z

′2d2y4z4d4 = v2z

′2d2x3v3d3.

Because of (5.6) we obtain the following equality in S:

xw = xw′,

where w = x1y1y4d1d4z′4 and w′ = x3v2v3d2d3z

′2. As S is cancellative, we

get that w = w′ in S.Note that w and w′ are different words. Indeed, suppose the contrary.

Then, since supp(x1)∩supp(x3) = ∅ and supp(z′2)∩supp(z′4) = ∅, we obtainthat supp(w) = supp(w′) = ∅. So, x1 = x3 (= x) and z2 = z4 (= z) and



the defining relations are x1 = z2 and x3 = z4. So, they are identical, acontradiction. It follows that indeed the words w,w′ are different.

If z2 is a subword of z′4 then z′2 is a subword of z′4 (thus z′2 is the emptyword as supp(z′2) ∩ supp(z′4) = ∅) and therefore z4 is not a subword of z′2.Similarly, if z4 is a subword of z′2 then z′4 is empty and hence z2 is not asubword of z′4. Therefore, we have that either z2 is not a subword of z′4 orz4 is not a subword of z′2. By symmetry, we may assume the former.

Thus, z2 is not a subword of z′4. This implies that v2z2d2 is not asubword of w. Since x1 is a proper subword of x1, x1y1d1 is not a subwordof w. Furthermore, since z′4 is a proper subword of z4, y4z4d4 is not asubword of w. If x3v3d3 is not a subword of w then w cannot be rewrittenin S and so (as w and w′ are different words) w 6= w′ in S, a contradiction.Consequently, x3v3d3 is a subword of w. Then v3, d3 are empty and x3 is asubword of x1. Thus, x3 is empty and x3 = x, x1 = x1x3 = x2

3x′ (equality

of words), for some x′ ∈ X.Notice that v2z2d2, x3v3d3, x1y1d1 (the last one because x3 is empty) are

not subwords of w′. Therefore (and again because w and w′ are differentwords but they are equal as elements of S), y4z4d4 is a subword of w′.Then y4, d4 are empty and z4 is a subword of z′2. We thus get that z′4is empty, z2 = z4z

′2 and z′2 = z4z

′ for some z′ ∈ Z. Hence z2 = z4z4z′

(equality of words). Now the defining relations look like x3 = z4 (hencex3, z4 are nonempty) and x2

3x′y1d1 = v2z2d2. The latter can be rewritten as

x23x′y1d1 = v2z2d2 = v2(z4z4z′)d2 = v2x

23z′d2. Let k be the largest positive

integer so that xk3 divides the word x′ and zk

4 divides the word z′. Writexk

3x0 = x′ and zk4z0 = z′ for some words x0, z0. Since S is cancellative, it

follows that xk+23 x′y1d1 = v2x

k+23 z′d2 and thus x0y1d1 = v2z0d2 holds in

S. Now it is clear that all wi, i = 1, 2, 3, 4, are not subwords of one of thewords x0y1d1, v2z0d2 (the former could only have w3 = x3 as a subword andthe latter only w4 = z4, but this would contradict the maximality of k).Therefore, this word cannot be rewritten in S and hence x0y1d1 = v2z0d2

as words. Because supp(x0y1d1)∩ supp(v2z0d2) = ∅, this implies that bothwords are empty. So the defining relations are x3 = z4 and xk+2

3 = zk+24 .

Therefore the relations are dependent, again a contradiction.Summarizing, we have shown that if supp(x1) ∩ supp(x3) 6= ∅, then

supp(z2) ∩ supp(z4) = ∅. By symmetry, if supp(z2) ∩ supp(z4) 6= ∅, thensupp(x1) ∩ supp(x3) = ∅. Interchanging the left and right hand side of thesecond equation in (5.5), it also follows that if supp(y1)∩supp(y4) 6= ∅ thensupp(v2) ∩ supp(v3) = ∅ and if supp(v2) ∩ supp(v3) 6= ∅ then supp(y1) ∩supp(y4) = ∅. The result therefore follows.



We note that, in the above lemma, if one of the words wi is empty, sayw1, then the elements in the support of w2 are invertible in S. Therefore,as S is a maximal order if and only if S/U(S) is a maximal order, this caseis reduced to one-relator monoids.

Theorem 5.2.2. Let S = 〈u1, . . . , un〉 be a finitely presented abelian monoidwith independent defining relations w1 = w2 and w3 = w4 and, for all i,|supp(wi)| ≥ 1. Then the semigroup S is a maximal order in its torsionfree group of quotients (or equivalently, by Theorem 5.0.1., the semigroupalgebra K[S] is an integrally closed domain) if and only if the followingconditions hold:

1. supp(w1) ∩ supp(w2) = ∅, supp(w3) ∩ supp(w4) = ∅,

2. Hsupp(w1) = ∅ or Hsupp(w2) = ∅,

3. Hsupp(w3) = ∅ or Hsupp(w4) = ∅,

4. if there exist i ∈ {1, 2}, j ∈ {3, 4} such that supp(wi) ∩ supp(wj) 6=∅, then one of the following properties holds (we may assume forsimplicity that i = 1 and j = 3):

• supp(wk) ∩ supp(wl) = ∅ for all pairs {k, l} 6= {1, 3} with k 6= l,and Hsupp(w2) = ∅ or Hsupp(w4) = ∅,

• there exists a pair k 6= l such that {2, 4} 6= {k, l} 6= {1, 3} andsupp(wk) ∩ supp(wl) 6= ∅ (for simplicity assume k = 2, l = 3),supp(w4) ∩ supp(wi) = ∅ for i = 1, 2, 3 and Hsupp(w4) = ∅.

Proof. Note that S = S1×S2, where S2 is the free abelian monoid generatedby

{u1, . . . , un} \ (4⋃

i=1

supp(wi))

and

S1 = 〈4⋃

i=1

supp(wi)〉.

Since S2 is a maximal order, it follows that S is a maximal order if andonly if S1 is a maximal order, i.e. we may assume that {u1, . . . , un} =⋃4

i=1 supp(wi).To prove the necessity of the conditions, suppose S is a cancellative

maximal order. The first property follows from Lemma 5.2.1. We prove



the second property by contradiction. So, assume that Hsupp(w1) 6= ∅ andHsupp(w2) 6= ∅. Note that there are two types of minimal primes in S.First, there are

Q = (ui, uj),

where ui and uj each belong to the support of different sides of one of thedefining relations and do not belong to the supports of the words in theother relation. To prove that Q is a prime ideal we may assume, by sym-metry, that ui, uj ∈ supp(w1) ∪ supp(w2). Clearly, S/Q is then generatedby the natural images of the elements uq, q 6= i, j, subject to the uniquerelation w3 = w4. Since ui, uj 6∈ supp(w3)∪ supp(w4), it is easily seen that(S/Q) \ {0} is a multiplicatively closed set, as desired. Second, there areminimal primes of the form

Q = (ui, uj , uk),

where ui belongs to the support of a word in each of the two relations,uj and uk belong to the support of a word in a defining relation but on adifferent side than ui, and furthermore uj and uk are involved in differentrelations. In particular, j 6= k by Lemma 5.2.1.

Choose ui ∈ Hsupp(w1), uj ∈ Hsupp(w2). We consider two cases.Case 1: ui or uj belongs to supp(w3)∪supp(w4). Without loss of gener-

ality we may assume that ui ∈ supp(w3) and supp(w4)∩ (⋃3

i=1 supp(wi)) =∅ (by Lemma 5.2.1). Take Q = (ui, uj , uk), uk ∈ supp(w4). Then Q is aminimal prime ideal of S and SQ = 〈ui, uj , uk〉U(SQ). Clearly, SQ/SQui

is generated by units and the natural images of uj and uk. The definingrelations of S take the following form in SQ/SQui:

0 = uαj v and 0 = ukw,

for some α ≥ 2, and v, w units. Hence the monoid SQ/SQui modulo itsunits is generated by uj subject to the unique relation uα

j = 0. There-fore (SQ/SQui) \ {0} is not a group and thus SQ 6= 〈ui〉U(SQ). Similarly,SQ/SQuj is generated by its units and the natural images of uj and uk.The defining relations of S take the following form in SQ/SQuj :

uβi v = 0 and uiw = ukx,

for some β ≥ 2, v, x units and w ∈ S. Hence, the monoid SQ/SQuj moduloits units is generated by {u1, . . . , un}\{uj , uk} subject to the unique relation

uβi = 0.



So also (SQ/SQuj) \ {0} is not a group and therefore SQ 6= U(SQ)〈uj〉.Because S is a maximal order and thus SQ is a valuation semigroup, itfollows that SQ = 〈uk〉U(SQ) and ukSQ is the unique prime ideal of SQ.We get that ui ∈ ukSQ. So SQ/ukSQ, modulo its units, is generated by uj

subject to the unique relation

0 = uαj ,

for some α ≥ 2. So, (SQ/SQuk) \ {0} also is not a group, again a contra-diction.

Case 2: ui, uj /∈ supp(w3) ∪ supp(w4). Then, Q = (ui, uj) is a minimalprime in S. Consider the localization SQ = 〈ui, uj〉U(SQ). Because ui ∈Hsupp(w1) and uj ∈ Hsupp(w2), by the same reasoning as above, we getSQ 6= 〈uj〉U(SQ), SQ 6= 〈ui〉U(SQ). Hence SQ 6= U(SQ)〈uq〉, for everygenerator uq of S, in contradiction with the fact that S is a maximal order.This finishes the proof of the second property, and thus, similarly, also ofthe third property.

To prove property four, it is sufficient to deal with defining relations

w1 = w2 and w3 = w4,

so that supp(w1) ∩ supp(w3) 6= ∅. Suppose furthermore that supp(w2) ∩supp(w3) 6= ∅. Note that Lemma 5.2.1 implies that

supp(w4) ∩ (3⋃

i=1

supp(wi)) = ∅.

We need to show that Hsupp(w4) = ∅. We prove this by contradiction, sosuppose Hsupp(w4) 6= ∅. Renumbering the generators, if necessary, we maywrite w4 = ual

l · · ·uann , for some l ≤ n, and, without loss of generality, one

can assume that an > 1. Consider the minimal prime Q = (ui, uj , un) ofS, with ui ∈ supp(w1) ∩ supp(w3) and uj ∈ supp(w2) ∩ supp(w3). So, thedefining relations are of the form,

uiv1 = ujv2 and uiujv3 = uall · · ·u

ann ,

for some v1, v2 and v3 in S. Then SQ = U(SQ)〈ui, uj , un〉. Since Hsupp(w1) =∅ or Hsupp(w2) = ∅, it is easily seen that ui ∈ U(SQ)〈uj〉 or uj ∈ U(SQ)〈ui〉.Say uj ∈ U(SQ)〈ui〉. Furthermore (by a reasoning as above and sincean > 1)

un /∈ U(SQ)〈ui〉.



So, since S is a maximal order, we must have SQ = U(SQ)〈un〉. HoweverSQ/SQun modulo its units is generated by ui subject to the unique relation

uγi = 0,

for some γ ≥ 2. So again (SQ/SQun) \ {0} is not a group, a contradiction.To finish the proof of the necessity, we need now to consider the case

where supp(w2)∩supp(w3) = supp(w2)∩supp(w4) = supp(w1)∩supp(w4) =∅. We show that Hsupp(w2) = ∅ or Hsupp(w4) = ∅. Assume the contrary,i.e., Hsupp(w2) 6= ∅, Hsupp(w4) 6= ∅ and Hsupp(w1) = Hsupp(w3) = ∅.Without loss of generality, we may assume un ∈ Hsupp(w4), so an > 1.Consider the minimal prime Q = (ui, uk, un) of S, where ui ∈ supp(w1) ∩supp(w3) and uk ∈ Hsupp(w2) (and thus ak > 1). The defining relationsare thus of the form

uiv1 = uakk v2 and uiv3 = ual

l · · ·uann .

for some v1, v2 and v3 in S. Then SQ = U(SQ)〈ui, uk, un〉. BecauseHsupp(w1) = Hsupp(w3) = ∅, it is easily seen that ui ∈ U(SQ)〈uk〉 andui ∈ U(SQ)〈un〉. With arguments as before we also see that uk /∈ U(SQ)〈un〉and un /∈ U(SQ)〈uk〉. Hence SQ is not of the form U(SQ)〈uq〉, for any gen-erator uq, a contradiction.

We now prove the sufficiency of the conditions. So, suppose that con-ditions (1),(2),(3) and (4) hold. If (supp(w1) ∪ supp(w2)) ∩ (supp(w3) ∪supp(w4)) = ∅ then S ∼= S1×S2, with S1 = 〈supp(w1)∪supp(w2) | w1 = w2〉and S2 = 〈supp(w3) ∪ supp(w4) | w3 = w4〉. Because the direct productof maximal orders again is a maximal order, it follows from Theorem 5.1.2that S is a maximal order. Hence, for the rest of the proof, we may assumethat condition (4) is not void.

We claim that if S is embedded in a group then the group SS−1 istorsion free (actually a free abelian group of rank n− 2). Indeed, becauseof the assumptions there exists ui and ε ∈ {1, 2} so that ui ∈ supp(wε)and Hsupp(wε) = ∅. Renumbering the generators, if necessary, we mayassume that i = 1. Then the relation w1 = w2 implies that u1 = wv−1

for some w, v ∈ S with supp(w) ∪ supp(v) ∪ {u1} = supp(w1) ∪ supp(w2),u1 6∈ supp(w) ∪ supp(v) and supp(w) ∩ supp(v) = ∅. It follows that

SS−1 = gr(u2, . . . , un | w3(wv−1, u2, . . . , un) = w4(wv−1, u2, . . . , un)).

If the second property of (4) holds then supp(w4)∩(⋃3

i=1 supp(wi)) = ∅ andHsupp(w4) = ∅. So, in particular, u1 6∈ supp(w4) and for uk ∈ supp(w4) we



have that uk 6∈ supp(w) ∪ supp(v) ∪ sup(w3) and

uk = w3(wv−1, u2, . . . , un)u−1

with w4 = uuk and supp(w4) = supp(u) ∪ {uk}. Hence we obtain thatSS−1 = gr({u2, . . . , un} \ {uk}) and this is a free abelian group of rankn−2, as claimed. If, on the other hand, the first property of (4) holds then,without loss of generality, we may assume that supp(w1) ∩ supp(w3) 6= ∅,Hsupp(w2) = ∅ and u1 ∈ supp(w2). So, u1 6∈ supp(w3). If Hsupp(w3) = ∅then choose uk ∈ supp(w3) and write w3 = ukv

′ with uk 6∈ supp(v′) andsupp(w3) = {uk}∪ supp(v′). So uk = w4(v′)−1. Note that u1 6∈ supp(w4)∪supp(v′). It follows that SS−1 = gr({u2, . . . , un} \ {uk}), a free abeliangroup of rank n − 2. Finally, if Hsupp(w3) 6= ∅ then Hsupp(w4) = ∅. Inthis case write w4 = ulv

′′ for some v′′ with ul 6∈ supp(v′′) and supp(w4) ={uk} ∪ supp(v′′). It follows that SS−1 = gr({u2, . . . , un} \ {ul}), again afree abelian group of rank n− 2, as desired.

So now we show that S is cancellative. By symmetry we can assumethat Hsupp(w4) = ∅. Then write

w2 = yγ11 · · · yγq

q , w4 = x1 · · ·xp−1xp,

γi ≥ 1, where x1, . . . , xp, y1, . . . , yq ∈ {u1, . . . , un}, and supp(w4) does notintersect nontrivially the support of any other word in the defining relations.

Let F be the free abelian monoid with basis supp(w1) ∪ {y1, . . . , yq} ∪supp(w3) ∪ {x1, . . . , xp−1}. Then let T = F/ρ, where ρ is the congruencedefined by the relation w1 = w2. Since Hsupp(w1) = ∅ or Hsupp(w2) = ∅,we know from Theorem 5.1.2 that T is a cancellative maximal order. Inparticular, TT−1 is a torsion free group. Consider the semigroup morphism

f : T × 〈u〉 −→ TT−1

defined by f(t) = t, for t ∈ T and f(u) = w3z−1 and z = x1 · · ·xp−1. Note

that f(w3) = f(zu). Hence the above morphism induces the followingnatural morphisms

T × 〈u〉 π−→ (T × 〈u〉)/ν f−→ TT−1,

with ν the congruence defined by the relation

w3 = zu.

Put M = (T × 〈u〉)/ν and note that

M ∼= S.



For simplicity we denote π(t) as t, for t ∈ T × 〈u〉. We note that π|T , therestriction of π to T , is injective. Indeed, suppose s, t ∈ T are such thatπ(s) = π(t). Then

s− t ∈ K[T × 〈u〉](zu− w3),

an ideal in K[T × 〈u〉]. So, s − t = α(zu − w3), for some α ∈ K[T × 〈u〉].Now K[T ×〈u〉] has a natural N-gradation, with respect to the degree in u.Clearly, s− t and w3 have degree zero. Let αh be the highest degree termof α with respect to this gradation. Then,

0 = αhzu.

Since T ×〈u〉 is contained in a torsion free group, we know that K[T ×〈u〉]is a domain. So we get that αh = 0 and thus α = 0. Hence s = t andtherefore indeed π|T is injective. So we will identify the element π(t) witht, for t ∈ T .

Next we note that u is a cancellable element in M . Indeed, let x, y ∈Mand suppose u x = u y. This means that

ux− uy ∈ K[T × 〈u〉](uz − w3),

i.e.

ux− uy = α(uz − w3) (5.8)

for some α ∈ K[T × 〈u〉], where x, y ∈ T × 〈u〉 are inverse images of x, y.Again consider the N-gradation on K[T × 〈u〉] via the degree in u. Let α0

be the zero degree component of α. Then it follows that

0 = α0w3.

Hence α0 = 0, as K[T ] is a domain, and thus

α ∈ K[T × 〈u〉]u.

Using again that K[T × 〈u〉] is a domain, we get from (5.8) that

x− y ∈ K[T × 〈u〉](uz − w3).

Hence x = y ∈M , as desired.In the above we thus have shown that u is cancellable in M . Hence

xp is cancellable in S. The argument of the proof holds for all elementsx1, . . . , xp. So, all elements x1, . . . , xp are cancellable in S. By a similar



argument, if Hsupp(w2) = ∅, this also holds for all elements yi ∈ supp(w2)\supp(w3).

On the other hand, if Hsupp(w2) 6= ∅ and thus Hsupp(w1) = ∅, thensimilarly one shows that ui is cancellable in S, for every ui ∈ supp(w1) \supp(w3). Clearly, S is contained in its localization SC , with respect to themultiplicatively closed set of the cancellable elements. In view of the formof the defining relations of S, this implies that SC is a group. So S, is acancellative monoid.

Finally, it remains to show that S is a maximal order in SS−1. Soassume that S satisfies conditions (1),(2),(3) and one of the properties in(4). Namely,

w1 = w2 and w3 = w4,

withsupp(w1) ∩ supp(w3) 6= ∅.

Furthermore, after renumbering if necessary, we may assume that

supp(w4) ∩ (3⋃

i=1

supp(wi)) = ∅

andHsupp(w1) = Hsupp(w4) = ∅.

Hence we can write the defining relations of S as follows:

u1 · · ·uk1uk2+1 · · ·uk3 = uak1+1

k1+1 · · ·uak2k2u

ak3+1

k3+1 · · ·uak4k4

ua11 · · ·uak1

k1u

bk1+1

k1+1 · · ·ubk2k2u

ak4+1

k4+1 · · ·uak5k5

= uk5+1 · · ·un,

with k1 ≤ k2 ≤ k3 ≤ k4 ≤ k5 < n (we agree that if k1 = k2, k2 = k3, k3 =k4 or k4 = k5 then the factors u

ak1+1

k1+1 · · ·uak2k2

, ubk1+1

k1+1 · · ·ubk2k2

, uk2+1 · · ·uk3 ,u

ak3+1

k3+1 · · ·uak4k4

, or uak4+1

k4+1 · · ·uak5k5

are the empty words).As said before, it is easily seen that the minimal prime ideals of S are

either of the formQ = (ui, uj),

where ui and uj each belong to the support of different sides of one of thedefining relations and do not belong to the supports of the words in theother relation, or of the form

Q = (ui, uj , uk),



where ui belongs to the support of a word in each of the two relations,uj and uk belong to the support of a word in a defining relation but on adifferent side than ui, and furthermore uj and uk are involved in differentrelations. Because of the defining relations in S, it is easy to see thatSQ = U(SQ)〈ux〉 for some x and this is a discrete valuation semigroup.Hence, to show that S is a maximal order, it is sufficient to show thatS =

⋂Q SQ, where Q runs through all minimal prime ideals of S. In order

to prove this, let s ∈⋂

Q SQ and write s =

ue11 . . . u

ek1k1u

ek1+1

k1+1 . . . uek2k2u

ek2+1

k2+1 . . . uek3k3u

ek3+1

k3+1 . . . uek4k4u

ek4+1

k4+1 . . . uek5k5u

ek5+1

k5+1 . . . uenn

for some ej ∈ Z. Note that, since the group SS−1 is free generated by

{u2, . . . , un−1},

we can assume that e1 = en = 0. We introduce the following three sets(and agree that max(∅) = 0):

A = {−ez | z ∈ {k5 + 1, . . . , n} with ez < 0},B = {−jl | jl = el − al max(A), l ∈ {1, . . . , k1} with jl < 0},C = {−em | m ∈ {k2 + 1, . . . , k3} with em < 0}.

Note that max(A), max(B), max(B ∪C) ≥ 0. We put jl = el − al max(A)for every l ∈ {1, . . . , k1}. To prove that s ∈ S it is sufficient to show thatthe following properties hold:

(n1) for every v ∈ {k4 + 1, . . . , k5}: ev ≥ av max(A),

(n2) for every w ∈ {k3 + 1, . . . , k4}: ew ≥ aw max(B ∪ C),

(n3) for every x ∈ {k1 + 1, . . . , k2}: ex ≥ ax max(B ∪ C) + bx max(A),

Indeed, if (n1),(n2) and (n3) hold then

ev = av max(A) + αv, for some αv ≥ 0, v ∈ {k4 + 1, . . . , k5},ew = aw max(B ∪ C) + αw, for some αw ≥ 0, w ∈ {k3 + 1, . . . , k4},em = −max(C) + αm, for some αm ≥ 0, m ∈ {k2 + 1, . . . , k3},ez = −max(A) + αz, for some αz ≥ 0, z ∈ {k5 + 1, . . . , n}.ex = ax max(B ∪ C) + bx max(A) + αx, for some αx ≥ 0, x ∈ {k1 + 1, . . . , k2}.



So, applying the first of the defining relations, we get

s = ua1 max(A)+j11 · · ·uak1

max(A)+jk1k1

uak1+1 max(B∪C)+bk1+1 max(A)+αk1+1

k1+1 · · ·uak2max(B∪C)+bk2

max(A)+αk2k2

u−max(C)+αk2+1

k2+1 · · ·u−max(C)+αk3k3

uak3+1 max(B∪C)+αk3+1

k3+1 · · ·uak4max(B∪C)+αk4

k4

uak4+1 max(A)+αk4+1

k4+1 · · ·uak5max(A)+αk5

k5u−max(A)+αk5+1

k5+1 · · ·u−max(A)+αnn

= uj11 · · ·ujk1

k1(uk2+1 · · ·uk3)

−max(C)(uak1+1

k1+1 · · ·uak2k2u

ak3+1

k3+1 · · ·uak4k4

)max(B∪C)

(uk5+1 · · ·un)−max(A)(ua11 · · ·uak1

k1u

bk1+1

k1+1 · · ·ubk2k2u

ak4+1

k4+1 · · ·uak5k5

)max(A)

uαk1+1

k1+1 · · ·uαk2k2

uαk2+1

k2+1 · · ·uαk3k3

uαk3+1

k3+1 · · ·uαk4k4

uαk4+1

k4+1 · · ·uαk5k5

uαk5+1

k5+1 · · ·uαnn

= uj11 · · ·ujk1

k1(uk2+1 · · ·uk3)

−max(C)(u1 · · ·uk1uk2+1 · · ·uk3)max(B∪C)

(uk5+1 · · ·un)−max(A)(ua11 · · ·uak1

k1u

bk1+1

k1+1 · · ·ubk2k2u

ak4+1

k4+1 · · ·uak5k5

)max(A)

uαk1+1

k1+1 · · ·uαk2k2

uαk2+1

k2+1 · · ·uαk3k3

uαk3+1

k3+1 · · ·uαk4k4

uαk4+1

k4+1 · · ·uαk5k5

uαk5+1

k5+1 · · ·uαnn

= uj11 · · ·ujk1

k1(u1 · · ·uk1)

max(B∪C)(uk2+1 · · ·uk3)max (B∪C)−max(C)

uαk1+1

k1+1 · · ·uαk2k2

uαk2+1

k2+1 · · ·uαk3k3

uαk3+1

k3+1 · · ·uαk4k4

uαk4+1

k4+1 · · ·uαk5k5

uαk5+1

k5+1 · · ·uαnn

= uj1+max(B∪C)1 · · ·ujk1

+max(B∪C)

k1(uk2+1 · · ·uk3)

max (B∪C)−max(C)

uαk1+1

k1+1 · · ·uαk2k2

uαk2+1

k2+1 · · ·uαk3k3

uαk3+1

k3+1 · · ·uαk4k4

uαk4+1

k4+1 · · ·uαk5k5

uαk5+1

k5+1 · · ·uαnn .

Since −max(C)+max(B∪C) ≥ 0 and jl +max(B∪C) ≥ 0 for 1 ≤ l ≤k1, it follows that s ∈ S, as desired. Indeed, if jl ≥ 0, then this is clear asmax(B∪C) ≥ 0. If jl < 0, then −jl ∈ B and max(B∪C) ≥ max(B) ≥ −jl.

We now prove conditions (n1), (n2) and (n3).

(n1) Suppose {k4 + 1, . . . , k5} 6= ∅ and fix some v ∈ {k4 + 1, . . . , k5}. Letz ∈ {k5+1, . . . , n}. Consider the minimal primeQ = (uv, uz). ClearlySQ = U(SQ)〈uv〉 and U(SQ) = gr({u1, . . . , un} \ {u1, uv, uz}). Also

s = s′uevv u

ezz = s′′uev+avez

v ∈ SQ = U(SQ)〈uv〉

for some s′, s′′ ∈ U(SQ). Since SS−1 is a free group with basis{u1, . . . , un}\{u1, uz}, it follows that ev +avez ≥ 0. Hence ev ≥ (−ez)av for everyz ∈ {k5 + 1, . . . , n} and, in particular, if z = n, then it follows thatev ≥ 0. Thus ev ≥ max(A)av, as desired.



(n2) Let w ∈ {k3 + 1, . . . , k4}. If {k2 + 1, . . . , k3} 6= ∅ and m ∈ {k2 +1, . . . , k3} then consider the minimal prime Q = (um, uw). ThenSQ = U(SQ)〈uw〉 with U(SQ) = gr({u1, . . . , un} \ {um, uw, un}). Fur-thermore,

s = s′uemm uew

w = s′′uew+awemw ∈ SQ = U(SQ)〈uw〉,

for some s′, s′′ ∈ U(SQ). Since SS−1 is a free group {u1, . . . , un} \{um, un}, it follows that ew + awem ≥ 0, and thus ew ≥ aw(−em) forevery m ∈ {k2 + 1, . . . , k3}.On the other hand, ifm ∈ {1, . . . , k1} and z ∈ {k5+1, . . . , n}, considerthe minimal prime Q = (um, uw, uz). Then SQ = U(SQ)〈uw〉 withU(SQ) = gr({u1, . . . , un} \ {um, uw, uz}). Furthermore,

s = s′uemm uew

w uezz = s′′uawem

w ueww uamez

m

= s′′′uew+aw(em+amez)w ∈ SQ = U(SQ)〈uw〉,

for some s′, s′′, s′′′ ∈ U(SQ). Since SS−1 is a free group with basis

{u1, . . . , un} \ {um, uz},

it follows that ew + aw(em + amez) ≥ 0. Hence ew ≥ aw(−em +am(−ez)), for every z ∈ {k5 + 1, . . . , n}, and thus ew ≥ aw(−jm),for every m ∈ {1, . . . , k1}. In particular, if z = n then we get ew ≥−awem. If additionally m = 1 then we get ew ≥ 0. Therefore, by thefirst part of the proof of (n2) it follows that ew ≥ aw max(B ∪C), asdesired.

(n3) Let x ∈ {k1 + 1, . . . , k2}. If {k2 + 1, . . . , k3} 6= ∅ and m ∈ {k2 +1, . . . , k3}, consider the minimal prime Q = (um, ux, uz) for z ∈ {k5 +1, . . . , n}. Then SQ = U(SQ)〈ux〉 with U(SQ) = gr({u1, . . . , un} \{um, ux, uz}). Furthermore,

s = s′uemm uex

x uezz = s′′uaxem

x uexx u

bxezx = s′′uaxem+ex+bxez

x ∈ SQ,

for some s′, s′′ ∈ U(SQ). Since SS−1 is a free group with basis

{u1, . . . , un} \ {um, uz},

it follows that axem + ex + bxez ≥ 0. Hence ex ≥ ax(−em) + bx(−ez).



On the other hand, ifm ∈ {1, . . . , k1} and z ∈ {k5+1, . . . , n}, considerthe minimal prime Q = (um, ux, uz). Then SQ = U(SQ)〈ux〉 withU(SQ) = gr({u1, . . . , un} \ {um, ux, uz}). Furthermore,

s = s′uemm uex

x uezz = s′′uaxem

x uexx u

amezm ubxez

x = s′′′uex+ax(em+amez)+bxezx ∈ SQ,

for some s′, s′′, s′′′ ∈ U(SQ). Since SS−1 is a free group with basis

{u1, . . . , un} \ {um, uz},

it follows that ex + ax(em + amez) + bxez ≥ 0. Hence ex ≥ ax(−em +am(−ez)) + bx(−ez), for every z ∈ {k5 + 1, . . . , n}, and thus ex ≥ax(−jm) + bx(−ez), for every m ∈ {1, . . . , k1}. In particular, if z = nthen we get ex ≥ ax(−em). If additionally m = 1 then we get ex ≥ 0.Therefore, by the first part of the proof of (n3), it follows that ex ≥ax max(B ∪ C) + bx max(A), as desired.

This ends the proof of the fact that S is a maximal order.

From Theorem 5.2.2, it follows that the abelian monoid S = 〈u1, · · · , u5〉,defined by the relations u1u2 = u3u4, u1u3 = u2

5 is not a maximal order.On the other hand, let S be the abelian monoid 〈u1, · · · , u6〉 on 6 gene-

rators defined by the relations u1u2 = u3u4, u1u2 = u5u6. Theorem 5.2.2yields that this monoid is an abelian maximal order on two relations (aswas also showed in Example 4.3.1). It is not so difficult to prove that theclass group of this monoid is isomorphic to Z4, since this monoid has 8minimal primes and the torsion free rank of BB−1 equals 4.

We finally describe the class groups of any finitely presented abelianmaximal order on two relations. We use the same notation as in the proof ofTheorem 5.2.2. Namely, if (supp(w1)∪supp(w2))∩(supp(w3)∪supp(w4)) =∅ then S ∼= S1 × S2, with S1 = 〈supp(w1) ∪ supp(w2) | w1 = w2〉 andS2 = 〈supp(w3) ∪ supp(w4) | w3 = w4〉. Clearly, in this case

cl(S) ∼= cl(S1)× cl(S2),

and the result follows from Theorem 5.1.4. Furthermore, if S satisfies oneof the properties in condition (4) in Theorem 5.2.2, we can write

S = 〈u1, . . . , un〉

with relations

u1 · · ·uk1uk2+1 · · ·uk3 = uak1+1

k1+1 · · ·uak2k2u

ak3+1

k3+1 · · ·uak4k4

ua11 · · ·uak1

k1u

bk1+1

k1+1 · · ·ubk2k2u

ak4+1

k4+1 · · ·uak5k5

= uk5+1 · · ·un,



with k1 ≤ k2 ≤ k3 ≤ k4 ≤ k5 < n (we agree that if k1 = k2, k2 = k3, k3 =k4 or k4 = k5 then the factors u

ak1+1

k1+1 · · ·uak2k2

, ubk1+1

k1+1 · · ·ubk2k2

, uk2+1 · · ·uk3 ,u

ak3+1

k3+1 · · ·uak4k4

, or uak4+1

k4+1 · · ·uak5k5

are the empty words).In the next lemma, we describe the principal ideals as divisorial products

of minimal prime ideals.

Lemma 5.2.3. Let

S = 〈u1, . . . , un | u1 · · ·uk1uk2+1 · · ·uk3 = uak1+1

k1+1 · · ·uak2k2u

ak3+1

k3+1 · · ·uak4k4

ua11 · · ·uak1

k1u

bk1+1

k1+1 · · ·ubk2k2u

ak4+1

k4+1 · · ·uak5k5

= uk5+1 · · ·un〉,

with k1 ≤ k2 ≤ k3 ≤ k4 ≤ k5 < n, be an abelian maximal order. Put Py,z,the minimal prime ideal of S generated by {uy, uz}, y ∈ {k2 + 1, . . . , k3},z ∈ {k3 + 1, . . . , k4} or y ∈ {k4 + 1, . . . , k5}, z ∈ {k5 + 1, . . . , n} andput Pt,v,x, the minimal prime ideal of S that is generated by {ut, uv, ux},t ∈ {1, . . . , k1}, v ∈ {k1 + 1, . . . , k2, k3 + 1, . . . , k4}, x ∈ {k5 + 1, . . . , n} ort ∈ {k2 + 1, . . . , k3}, v ∈ {k1 + 1, . . . , k2}, x ∈ {k5 + 1, . . . , n}. Then

1. for every w ∈ {k1 + 1, . . . , k2},

Suw =

n∏l=k5+1

k1∏m=1

Pm,w,l

k3∏m=k2+1

Pm,w,l

∗

2. for every w ∈ {k3 + 1, . . . , k4},

Suw =

n∏l=k5+1

(k1∏

m=1

Pm,w,l

)k3∏

m=k2+1

Pm,w

∗

3. for every w ∈ {k4 + 1, . . . , k5},

Suw =

n∏l=k5+1

Pw,l

∗

4. for every w ∈ {1, . . . , k1},

Suw =

n∏l=k5+1

k2∏m=k1+1

P amw,m,l

k4∏m=k3+1

P amw,m,l

∗



5. for every w ∈ {k2 + 1, . . . , k3},

Suw =

n∏l=k5+1

k2∏m=k1+1

P amw,m,l

k4∏m=k3+1

P amw,m

∗

6. for every w ∈ {k5 + 1, . . . , n},

Suw =

k1∏l=1

k2∏m=k1+1

P aml,m,w

k4∏m=k3+1

P aml,m,w

al∗

∗

k2∏m=k1+1

k1∏l=1

Pl,m,w

k3∏l=k2+1

Pl,m,w

bmk5∏

l=k4+1

P all,w

∗

.

Proof. Note that it follows from the proof of Theorem 5.2.2 that the mini-mal primes of S are as described in the statement of the lemma. We onlywill prove statements four and six. To prove the former, let w ∈ {1, . . . , k1}.Then,

(uw, uk1+1)ak1+1 ∗ · · · ∗ (uw, uk2)ak2 ∗ (uw, uk3+1)ak3+1 ∗ · · · ∗ (uw, uk4)

ak4

= ((uw, uk1+1)ak1+1 · · · (uw, uk2)ak2 (uw, uk3+1)ak3+1 · · · (uw, uk4)

ak4 )∗

= (S : (S : (uw, uk1+1)ak1+1 · · · (uw, uk2)ak2 (uw, uk3+1)ak3+1 · · · (uw, uk4)

ak4 )) .

Because

uak+1

k1+1 · · ·uak2k2u

ak3+1

k3+1 · · ·uak4k4

= u1 · · ·uw · · ·uk1uk2+1 · · ·uk3

∈ (uw, uk1+1)ak1+1 · · · (uw, uk2)ak2 (uw, uk3+1)ak3+1 · · · (uw, uk4)

ak4 ,

it easily is verified that

(uw, uk1+1)ak1+1 · · · (uw, uk2)ak2 (uw, uk3+1)ak3+1 · · · (uw, uk4)

ak4 ⊆ Suw,

and hence

(S : (S : (uw, uk1+1)ak1+1 · · · (uw, uk2)ak2 (uw, uk3+1)ak3+1 · · · (uw, uk4)

ak4 )) ⊆ Suw.

On the other hand,

uak1+1+···+ak2

+ak3+1+···+ak4w


ak4 ,



implies that

(S : (uw, uk1+1)ak1+1 · · · (uw, uk2)ak2 (uw, uk3+1)ak3+1 · · · (uw, uk4)

ak4 ) ⊆ Su−lw ,

for some positive integer l. We claim that one may take l = 1. We showthis by contradiction. So suppose there exists an element g in


ak4 ) ⊆ Su−lw

such that g /∈ Su−1w . Hence g = su−l′

w , with s /∈ Suw and l′ ≥ 2. Because

u1 · · ·uw · · ·uk1uk2+1 · · ·uk3 = uak1+1

k1+1 · · ·uak2k2u

ak3+1

k3+1 · · ·uak4k4


ak4 ,

we get that

su−l′w u1 · · ·uw · · ·uk1uk2+1 · · ·uk3 = su−l′+1

w t ∈ S,

where t = u1 · · ·uw−1uw+1 · · ·uk1uk2+1 · · ·uk3 . Consequently, st ∈ Sul′−1w .

Since s /∈ Suw, the defining relations imply that st /∈ Suw and hencel′−1 = 0, a contradiction. This proves the claim. Therefore we obtain that


ak4 ) ⊆ Su−1w

and hence Suw ⊆

(S : (S : (uw, uk1+1)ak1+1 · · · (uw, uk2)ak2 (uw, uk3+1)ak3+1 · · · (uw, uk4)

ak4 )) .

So we have shown that

Suw =

k2∏m=k1+1

(uw, um)am

k4∏m=k3+1

(uw, um)am

∗

. (5.9)

Let v ∈ {k1 + 1, . . . , k2, k3 + 1, . . . k4}. Because Pw,v,k5+1, . . . , Pw,v,n aredifferent minimal primes,

Pw,v,k5+1 ∗ · · · ∗ Pw,v,n = Pw,v,k5+1 ∩ · · · ∩ Pw,v,n.

As the intersection consists of elements that are either products of genera-tors with uw or uv involved or the full product

uk5+1 · · ·un = ua11 · · ·uaw

w · · ·uak1k1u

bk1+1

k1+1 · · ·ubk2k2u

ak4+1

k4+1 · · ·uak5k5,



we get that

(uw, uv) =

n∏l=k5+1

Pw,v,l

∗

. (5.10)

From (5.9) and (5.10) one obtains statement four of the lemma.To prove the sixth statement of the lemma, let w ∈ {k5+1, . . . , n}. One

readily verifies that

Suw =

k1∏l=1

(ul, uw)al

k2∏m=k1+1

(um, uw)bm

k5∏l=k4+1

(ul, uw)al

∗

. (5.11)

For v ∈ {1, . . . , k1}, one also obtains that

(uv, uw) =

k2∏m=k1+1

(Pv,m,w)am

k4∏m=k3+1

(Pv,m,w)am

∗

. (5.12)

Furthermore, for v′ ∈ {k1 + 1, . . . , k2}, one can easily prove that

(uv′ , uw) =

k1∏l=1

Pl,v′,w

k3∏l=k2+1

Pl,v′,w

∗

. (5.13)

From (5.11), (5.12) and (5.13) we get

Suw =

k1∏l=1

k2∏m=k1+1

P aml,m,w

k4∏m=k3+1

P aml,m,w

al∗

∗

k2∏m=k1+1

k1∏l=1

Pl,m,w

k3∏l=k2+1

Pl,m,w

bmk5∏

l=k4+1

P all,w

∗

.

So statement six also has been proved.

Theorem 5.2.4. Let

S = 〈u1, . . . , un | u1 · · ·uk1uk2+1 · · ·uk3 = uak1+1

k1+1 · · ·uak2k2u

ak3+1

k3+1 · · ·uak4k4

ua11 · · ·uak1

k1u

bk1+1

k1+1 · · ·ubk2k2u

ak4+1

k4+1 · · ·uak5k5

= uk5+1 · · ·un〉



with k1 ≤ k2 ≤ k3 ≤ k4 ≤ k5 < n be an abelian maximal order. Then

cl(K[S]) ∼= cl(S) ∼= Zf × (Zd1)k1+k3−k2−1 × (Zd2)

n−k5−1,

where

f = (k3 − k2)(k4 − k3) + (k5 − k4)(n− k5) + k1(k4 − k3 + k2 − k1)(n− k5)+ (k3 − k2)(k2 − k1)(n− k5)− (n− 2),

withd1 = gcd(ak1+1, . . . , ak2 , ak3+1, . . . , ak4)

andd2 = gcd(a1d1, . . . , ak1d1, bk1+1, . . . , bk2 , ak4+1, . . . , ak5).

In particular, if d1 = d2 = 1, then the class group of S is torsion free.

Proof. It is shown in the proof of Theorem 5.2.2 that SS−1 ∼= Fan−2, thefree abelian group of rank n − 2. Because U(S) = {1}, we get that thetorsion free rank of P (S) equals the torsion free rank of SS−1. Since thetorsion free rank of cl(S) is the difference of the torsion free rank of D(S)and the torsion free rank of P (S), to establish the description of the torsionfree part of cl(S), we only need to show that there are (k3− k2)(k4− k3) +(k5−k4)(n−k5)+k1(k4−k3 +k2−k1)(n−k5)+ (k3−k2)(k2−k1)(n−k5)elements in X1(S). But this easily follows from the description of theminimal primes given in the proof of Theorem 5.2.2.

Clearly,

P (S) = gr(Suw | w ∈ {1, . . . , k1} ∪ {k1 + 1, . . . , k2} ∪ {k2 + 1, . . . , k3}∪ {k3 + 1, . . . , k4} ∪ {k4 + 1, . . . , k5} ∪ {k5 + 1, . . . , n})

and thus

cl(S) = gr(P | P ∈ X1(S))/ gr(Suw | w ∈ {1, . . . , k1}∪ {k1 + 1, . . . , k2} ∪ {k2 + 1, . . . , k3} ∪ {k3 + 1, . . . , k4}∪ {k4 + 1, . . . , k5} ∪ {k5 + 1, . . . , n}).

Furthermore, by using relations (1) and (2) from Lemma 5.2.3, we canrewrite in cl(S)

P1,w,k5+1 =

n∏m=k5+2

P−11,w,m

k1∏l=2

n∏m=k5+1

P−1l,w,m

k3∏l=k2+1

n∏m=k5+1

P−1l,w,m

∗



for every w ∈ {k1 + 1, . . . , k2}, and

P1,w,k5+1 =

n∏m=k5+2

P−11,w,m

k1∏l=2

n∏m=k5+1

P−1l,w,m

k3∏l=k2+1

P−1l,w

∗

,

for every w ∈ {k3 + 1, . . . , k4}. Similarly, by using relation (3) fromLemma 5.2.3, in cl(S) we can rewrite

Pw,k5+1 =

n∏m=k5+2

P−1w,m

∗

for every w ∈ {k4 + 1, . . . , k5}.By relations (1)-(6) in Lemma 5.2.3, in D(S) we have: k1∏

l=1

Sul

k3∏l=k2+1

Sul

∗

=

k2∏l=k1+1

Suall

k4∏l=k3+1

Suall

∗

and n∏l=k5+1

Sul

∗

=

k1∏l=1

Suall

k2∏l=k1+1

Subll

k5∏l=k4+1

Suall

∗

.

ThereforeSu1, Suk5+1 ∈ gr(Sui | i 6= 1, k5 + 1) ⊆ D(S).

This implies that cl(S) can be described as follows

gr(Pα | α ∈ A)/ gr(Suw | w ∈ {2, . . . , k1}∪{k2+1, . . . , k3}∪{k5+2, . . . , n})

where A = the set of all pairs y, z such that

y ∈ {k2 + 1, . . . , k3}, z ∈ {k3 + 1, . . . , k4}

ory ∈ {k4 + 1, . . . , k5}, z ∈ {k5 + 2, . . . , n}

and of all triples t, v, x such that

t ∈ {1, . . . , k1}, v ∈ {k1 + 1, . . . , k2, k3 + 1, . . . , k4}, x ∈ {k5 + 1, . . . , n}with (t, x) 6= (1, k5 + 1), or

t ∈ {k2 + 1, . . . , k3}, v ∈ {k1 + 1, . . . , k2}, x ∈ {k5 + 1, . . . , n}.



Indeed, factoring out the group P (S) in the presentation of cl(S) is equiv-alent with making elements listed in statements (1)-(6) in Lemma 5.2.3trivial. Namely, making the elements in first three statements of thislemma trivial yields the fact that we can limit the generators as formu-lated (by the rewriting). On the other hand, factoring out Suw, for w ∈{2, . . . , k1} ∪ {k2 + 1, . . . , k3} ∪ {k5 + 2, . . . , n}, is equivalent with makingthe elements of the remaining statements (4), (5) and (6) trivial.

Using again Lemma 5.2.3, we get that cl(S) can be described as

gr(Pα | α ∈ A)/ gr(Qd1w , S

d1w′ , R

d2w′′ | w ∈ {2, . . . , k1},

w′ ∈ {k2 + 1, . . . , k3}, w′′ ∈ {k5 + 2, . . . , n}),

where

Qw =

n∏l=k5+1

k2∏m=k1+1

Pαmw,m,l

k4∏m=k3+1

Pαmw,m,l

∗

and

Sw′ =

n∏l=k5+1

k2∏m=k1+1

Pαmw′,m,l

k4∏m=k3+1

Pαmw′,m

∗

with αmd1 = am, for m ∈ {k1 + 1, . . . , k2} ∪ {k3 + 1, . . . , k4} and with

d1 = gcd(ak1+1, . . . , ak2 , ak3+1, . . . , ak4).

Also

Rw′′ =

k1∏l=1

k2∏m=k1+1

Pβl,m

l,m,w′′

k4∏m=k3+1

Pβl,m

l,m,w′′

∗

∗

k2∏m=k1+1

k1∏l=1

Pl,m,w′′

k3∏l=k2+1

Pl,m,w′′

γmk5∏

l=k4+1

P γl

l,w′′

∗

=

k1∏l=1

k2∏m=k1+1

Pβl,m+γm

l,m,w′′

k4∏m=k3+1

Pβl,m

l,m,w′′

∗

∗

k2∏m=k1+1

k3∏l=k2+1

P γm

l,m,w′′

k5∏l=k4+1

P γl

l,w′′

∗

,



with βl,md2 = alam, for l ∈ {1, . . . , k1}, m ∈ {k1 + 1, . . . , k2} ∪ {k3 +1, . . . , k4}, γmd2 = bm, for m ∈ {k1 + 1, . . . , k2}, γld2 = al, for l ∈ {k4 +1, . . . , k5} and with

d2 = gcd(a1d1, . . . , ak1d1, bk1+1, . . . , bk2 , ak4+1, . . . , ak5)= gcd(a1{ak1+1, ..., ak2 , ak3+1, ..., ak4}, . . . , ak1{ak1+1, ..., ak2 , ak3+1, ..., ak4},

bk1+1, . . . , bk2 , ak4+1, . . . , ak5).

Consider the subgroup

H = gr(Qw, Sw′ , Rw′′ | w ∈ {2, . . . , k1},w′ ∈ {k2 + 1, . . . , k3}, w′′ ∈ {k5 + 2, . . . , n})

of the free abelian group

F = gr(Pα | α ∈ A)

with basis{Pα | α ∈ A}.

We claim that F/H is a torsion free group, and hence that cl(S) is torsionfree, provided that d1 = d2 = 1. In order to prove this, we need to showthat F/H does not contain elements of order p, for any prime p. Let Fp =Zp ⊗Z F , a Zp-vector space. Since we use the multiplicative notation forgroups, we note that the scalars of basis elements are written as exponents.

The natural image of f ∈ F in Fp is denoted by f . Clearly, the setconsisting of the elements P y,z and P t,v,x forms a basis of Fp. Let Hp =Zp⊗ZH, a subspace of Fp. The natural image of z ∈ Z in Zp we denote byz.

It is sufficient to prove that if f ∈ F with fp = h ∈ H then fp = (h′)p forsome h′ ∈ H. Indeed, since F is torsion free, we then get that f = h′ ∈ H.To prove the former, it is sufficient to show that the set

{Qw, Sw′ , Rw′′ | w ∈ {2, . . . , k1}, w′ ∈ {k2+1, . . . , k3}, w′′ ∈ {k5+2, . . . , n}}

is Zp-linearly independent. Indeed, let f ∈ F be so that fp ∈ H. Writefp =

∏w,w′,w′′ Qxw

w Sxw′w′ R

xw′′w′′ , with each xw, xw′ , xw′′ ∈ Z. Then 1 = f

p =∏w,w′,w′′ Q

xw

w Sxw′w′ R

xw′′w′′ . Because of the Zp-linear independence, we thus get

that all xw = xw′ = xw′′ = 0. Hence xw = pyw, xw′ = pyw′ and xw′′ = pyw′′

for some yw, yw′ , yw′′ ∈ Z. Consequently, fp = (h′)p with

h′ =∏

w,w′,w′′

Qyw

w Syw′w′ R

yw′′w′′ ,



as desired.In order to prove that {Qw, Sw′ , Rw′′ | w ∈ {2, . . . , k1}, w′ ∈ {k2 +

1, . . . , k3}, w′′ ∈ {k5 + 2, . . . , n}} is Zp-linearly independent, assume

δw, δw′ , δw′′ ∈ Zp

are such that f =∏

w,w′,w′′ Qδw

w Sδw′w′ R

δw′′w′′ = 1. We need to show that all

δw, δw′ , δw′′ are zero. Let w ∈ {2, . . . , k1}. For x ∈ {k1 + 1, . . . , k2},the Zp-exponent of Pw,x,k5+1 in f is αx δw. Hence, we get that αx δw = 0.Similarly, for y ∈ {k3+1, . . . , k4}, the exponent of Pw,y,k5+1 is αy δw. Henceαy δw = 0. Since p is prime, it follows that, for every w ∈ {2, . . . , k1}, eitherδw = 0 or

p | gcd(αk1+1, . . . , αk2 , αk3+1, . . . , αk4).

Because gcd(αk1+1, . . . , αk2 , αk3+1, . . . , αk4) = 1, we get that δw = 0, asdesired.

Now let w′ ∈ {k2 + 1, . . . , k3} and x ∈ {k1 + 1, . . . , k2}, y ∈ {k3 +1, . . . , k4}. The exponent of Pw′,x,k5+1, respectively Pw′,y, in f is αx δw′ ,respectively αy δw′ . Because

gcd(αk1+1, . . . , αk2 , αk3+1, . . . , αk4) = 1,

it follows that δw′ = 0, again as desired.So now δw = δw′ = 0 and

∏w′′ R

δw′′w′′ = 1. We have to prove that δw′′ = 0.

Therefore, let x ∈ {k2 + 1, . . . , k3}. The exponent of P x,y,w′′ is γy δw′′ , forevery y ∈ {k1 + 1, . . . , k2}. On the other hand, for x ∈ {1, . . . , k1}, theexponent of P x,y,w′′ is (βx,y + γy)δw′′ , for every y ∈ {k1 + 1, . . . , k2}. Hencethe former case implies that γy δw′′ = 0, for every y ∈ {k1 + 1, . . . , k2} andthus the latter case implies that βx,y δw′′ = 0, for every x ∈ {1, . . . , k1}. Onthe other hand, if x ∈ {1, . . . , k1} and z ∈ {k3 + 1, . . . , k4}, the exponentof P x,z,w′′ is βx,z δw′′ and thus βx,z δw′′ = 0. Finally, if z ∈ {k4 + 1, . . . , k5},the exponent of P z,w′′ is γz δw′′ and thus γz δw′′ = 0. Because

gcd ( βi,j , γl | i ∈ {1, . . . , k1}, j ∈ {k1 + 1, . . . , k2, k3 + 1, . . . , k4},l ∈ {k1 + 1, . . . , k2, k4 + 1, . . . , k5}) = 1,

we thus obtain that δw′′ = 0. This ends the proof of the fact that allδw, δw′ , δw′′ are zero, and hence F/H is torsion free, if d1 = d2 = 1.

We now consider the general case, that is d1 and d2 are not neces-sarily equal to 1. From the above it follows that the natural image of{Qw, Sw′ , Rw′′ | w ∈ {2, . . . , k1}, w′ ∈ {k2 + 1, . . . , k3}, w′′ ∈ {k5 +


5.3. Comments and problems 98

2, . . . , n}} in Fp is linearly independent for every prime p. Hence this setis Z-independent in the free abelian group F . We now show that in cl(S)

gr(Qd1w , S

d1w′ , R

d2w′′ | w ∈ {2, . . . , k1}, w′ ∈ {k2+1, . . . , k3}, w′′ ∈ {k5+2, . . . , n})

∼= (Zd1)k1+k3−k2−1 × (Zd2)

n−k5−1,

with gr(Qw) ∼= Zd1 , gr(Sw′) ∼= Zd1 and gr(Rw′′) ∼= Zd2 . For this it issufficient to show, in F , that∏

w,w′,w′′

Qδww S

δw′w′ R

δw′′w′′ ∈ gr(Qd1

w , Sd1w′ , R

d2w′′ | w ∈ {2, . . . , k1}, (5.14)

w′ ∈ {k2 + 1, . . . , k3}, w′′ ∈ {k5 + 2, . . . , n}),

with 0 ≤ δw, δw′ < d1 and 0 ≤ δw′′ < d2, implies

δw = δw′ = δw′′ = 0.

To prove the latter, note that, since {Qw, Sw′ , Rw′′ | w ∈ {2, . . . , k1}, w′ ∈{k2 + 1, . . . , k3}, w′′ ∈ {k5 + 2, . . . , n}} is a Z-linear independent set in F ,from (5.14) we get that, for each w, Qδw

w ∈ gr(Qd1w ). This clearly implies

that δw = 0. Similarly we get that δw′ = δw′′ = 0, and that finishes theproof of the theorem.

5.3 Comments and problems

In this final section we remark that Lemma 5.2.1 cannot be extended in anatural way to semigroups that are defined by more than two relations. Inorder to verify that the monoid is a maximal order, one of the propertiesto check first is whether the monoid is cancellative. In the case of monoidsthat are presented by at most two relations, we were able to show that atleast one word in the defining relations does not overlap with any otherword. Then, using also the maximal order condition, we obtained fullcontrol on the type of relations needed. Of course, there are many examplesof cancellative semigroups defined by more than two relations, where theabove mentioned property is not satisfied. It is unclear to the authorswhen such semigroups will be maximal orders. We illustrate this with thefollowing two examples, each defined via three relations, and every word inthe defining relations overlaps with at least one other word. However thefirst one is a maximal order while the second one is not.



Example 5.3.1. The abelian monoid

S = 〈u1, u2, u3, u4, u5 | u21 = u2u3, u1u4 = u2u5, u1u5 = u3u4〉

is a maximal order.

Proof. It is easy to see that

S = 〈u2, u3, u4〉 ∪ u1〈u2, u3, u4〉 ∪ 〈u3, u4, u5〉u5.

Let G = gr(x1, x2, x3, x4, x5 | x21 = x2x3, x1x4 = x2x5, x1x5 = x3x4).

Clearly, there is a natural homomorphism f of S toG. Also, G = gr(x1, x2, x4),a free abelian group of rank 3 and G = gr(x2, x3, x4)∪gr(x2, x3, x4)x1. AlsoG = gr(x3, x4, x5). Note that x5 = x−1

2 x1x4.We claim that f is injective, and thus that S is cancellative. For this

first note that gr(x2, x3, x4) also is free of rank 3. Hence to show thatf is injective it is sufficient to show that 〈x2, x3, x4〉 ∩ 〈x3, x4, x5〉x5 =∅ and x1〈x2, x3, x4〉 ∩ 〈x3, x4, x5〉x5 = ∅. To deal with the former, as-sume xα′

2 xβ′

3 xγ′

4 = xα3x

β4x

γ5x5, for some α′, β′, γ′, α, β, γ ≥ 0. Because G =

gr(x2, x3, x4) ∪ gr(x2, x3, x4)x1, it is easily seen that γ = 2δ + 1 for someδ ≥ 0. Hence

xα′2 x

β′

3 xγ′

4 = xα3x

β4x

2δ+15 x5

= xα3x

β4 (x−1

2 x4x1)2δ+1x−12 x4x1

= xα3x

β4x

−2δ−12 x2δ+1

4 x2δ+11 x−1

2 x4x1

= xα3x

β+2δ+24 x−2δ−2

2 (x21)

δ+1

= xα3x

β+2δ+24 x−2δ−2

2 (x2x3)δ+1

The independence of the generators x2, x3 and x4 implies that α′ = −δ−1 <0, a contradiction.

To deal with the second case, assume x1xα2x

β3x

γ4 = xα′

3 xβ′

4 xγ′

5 x5, forsome α′, β′, γ′, α, β, γ ≥ 0. Because G = gr(x2, x3, x4) ∪ gr(x2, x3, x4)x1, itis easily seen that γ′ = 2δ for some δ ≥ 0. Hence

x1xα2x

β3x

γ4 = xα′

3 xβ′

4 x2δ5 x5

= xα′3 x

β′

4 (x−12 x4x1)2δx−1

2 x4x1

= xα′3 x

β′

4 x−2δ2 x2δ

4 x2δ1 x

−12 x4x1

= xα′3 x

β′

4 x−2δ2 x2δ

4 (x2x3)δx−12 x4x1

The independence of the generators x2, x3 and x4 implies that α = −2δ +δ − 1 < 0, a contradiction.



So, indeed S is cancellative and thus G ∼= SS−1. Hence we may iden-tify G and SS−1, and thus also xi with ui. Next we show that S is amaximal order. For this we first list the minimal prime ideals Q of S:Q1 = (u1, u2, u3), Q2 = (u1, u2, u4), Q3 = (u1, u3, u5), Q4 = (u4, u5).It is readily verified that for any SQ = U(SQ)〈ui〉, for some i. Henceeach SQ is a discrete valuation semigroup. So it remains to show thatS = ∩QSQ. Therefore assume g = ua3

3 ua44 u

a55 ∈ ∩QSQ, with a3, a4, a5 ∈ Z.

Since u4, u5 ∈ U(SQ1),SQ1 = U(SQ1)〈u3〉,

and g = ua33 v1, for some v1 ∈ U(SQ1), we get that

a3 ≥ 0.

BecauseSQ2 = U(SQ2)〈u4〉

and u3, u5 ∈ U(SQ2) we obtain that g = ua44 v

′ for some v′ ∈ U(SQ2). Hence

a4 ≥ 0.

So g = ua33 u

a44 u

a55 , with a3, a4 ≥ 0. To show that g ∈ S it is sufficient to

show that a5 ≥ max(−2a3,−a4).Indeed, suppose that max(−2a3,−a4) = −2a3, so a5 ≥ −2a3 ≥ −a4.

Then

g = ua33 u

2a3+α44 ua5

5

= uα44 (u3u

24)

a3ua55

= uα44 u2a3+a5

5 ,

for some α4 ≥ 0 and clearly, by the above condition g ∈ S.On the other hand, if max(−2a3,−a4) = −a4, so a5 ≥ −a4 ≥ −2a3,

then

g = ua33 u

a44 u

−a4+α55

= ua33 u

a44 (u1u

−13 u−1

4 )a4uα55

= ua41 u

a3−a43 uα5

5 ,

for some α5 ≥ 0. If a4 is even,

g = ua41 u

a3−a43 uα5

5

= u2(a4/2)1 ua3−a4

3 uα55

= (u2u3)a4/2ua3−a43 uα5

5

= ua4/22 u

a3−a4/23 uα5

5 ,



and since 2a3 ≥ a4, this element clearly is in S.If a4 is odd, the condition 2a3 ≥ a4 is equivalent with 2a3 ≥ a4 + 1 and

g = ua41 u

a3−a43 uα5

5

= u2(

a4−12

)

1 u1ua3−a43 uα5

5

= (u2u3)a4−1

2 u1ua3−a43 uα5

5

= ua4−1

22 u1u

a3−a4+a4−1

23 uα5

5 ,

since 2a3 ≥ a4 + 1 is equivalent with a3− a4 + a4−12 ≥ 0, g is again in S, as

desired.It remains to show that, if g = ua3

3 ua44 u

a55 ∈ ∩SQ, then a5 ≥ max(−2a3,−a4).

Now, sinceSQ3 = gr(u3u

24u−25 , u4)〈u5〉,

elements of SQ3 are of the form

uβ23 u

2β24 u−2β2

5 uβ44 u

β55 ,

with β2, β4 ∈ Z and β5 ≥ 0. But then, since clearly, −2β2 + β5 + 2β2 ≥ 0,we get a5 + 2a3 ≥ 0.

On the other hand, since

SQ4 = gr(u3u4u−15 , u3u

24u−25 , u3)〈u4〉,

elements of SQ3 are of the form

uγ13 u

γ14 u

−γ15 uγ2

3 u2γ24 u−2γ2

5 uγ33 u

γ44 ,

with γ1, γ2, γ3 ∈ Z and γ4 ≥ 0. But then, since clearly, γ1 + 2γ2 + γ4 −γ1 − 2γ2 ≥ 0, we get a5 + a4 ≥ 0, as desired. So S is indeed a maximalorder.

Example 5.3.2. The abelian monoid

S = 〈u1, u2, u3, u4, u5, u6 | u21 = u3u4, u1u4 = u2u5, u

22 = u5u6〉

is cancellative, but not a maximal order.

Proof. We start by proving that the generators u3 and u6 are cancellableelements of S. Indeed, as in the proof of Theorem 5.2.2, write

S = T × 〈u6〉/ν,



with ν the congruence defined by the relation u22 = u5u6 and with

T = 〈u1, u2, u3, u4, u5 | u21 = u3u4, u1u4 = u2u5〉.

Write elements of S again as s with s ∈ T × 〈u6〉. Suppose

u6 w1 = u6 w2,

for some words w1 and w2 in S. Then

u6w1 − u6w2 ∈ K[T × 〈u6〉](u22 − u5u6),

i.e.

u6w1 − u6w2 = α(u22 − u5u6) (5.15)

for some α ∈ K[T × 〈u6〉], where w1, w2 ∈ T × 〈u〉 are inverse images ofw1, w2. Consider the N-gradation on K[T × 〈u6〉] via the degree in u6. Letα0 be the zero degree component of α. Then it follows that

0 = α0u22.

Hence α0 = 0, as K[T × 〈u6〉] is a domain, and thus

α ∈ K[T × 〈u6〉]u.

Using again that K[T × 〈u6〉] is a domain, we get from (5.15) that

w1 − w2 ∈ K[T × 〈u6〉](u22 − u5u6).

Hence w1 = w2 ∈ S, as desired.Similarly, one proves that u3 is cancellable by considering

S = T × 〈u3〉/ν,

with ν the congruence defined by the relation u21 = u3u4 and with

T = 〈u1, u2, u4, u5, u6 | u22 = u5u6, u1u4 = u2u5〉.

Hence we can localize S at 〈u3, u6〉, i.e. we can invert u3 and u6 and thus

S ⊆ 〈u1, u2, u3, u6, u−13 , u−1

6 | u1(u21u−13 ) = u2(u2

2u−16 )〉.

SoS ⊆ S′ = 〈u1, u2, u3, u6, u

−13 , u−1

6 | u31u6 = u3

2u3〉.



Now

S′ = 〈u2, u3, u6, u−13 , u−1

6 〉∪u1〈u2, u3, u6, u−13 , u−1

6 〉∪u21〈u2, u3, u6, u

−13 , u−1

6 〉

and clearly H = 〈u2, u3, u6, u−13 , u−1

6 〉 = 〈u2〉 gr(u3, u6) is a cancellativesemigroup in HH−1 = gr(u2, u3, u6). Clearly also HH−1 is a subgroup ofindex 3 in the group gr(u1, u2, u3, u6 | u3

1u6 = u32u3) = gr(u1, u2, u3), a free

abelian group of rank 3. So S′ and thus S is cancellative.That S is not a maximal order, is easily seen as Q = (u1, u2, u4, u5) is

a minimal prime of S so that SQ is not a discrete valuation semigroup.


Chapter 6

Examples of maximal orders

In this chapter we illustrate the theory developed in the preceding chapterson several concrete classes of monoids and their monoid algebras.

In [27] Gateva-Ivanova and Van den Bergh introduced a new class ofmonoids T , called monoids of I-type, with the aim of constructing non-commutative algebras that share many properties with polynomial algebrasin finitely many commuting variables. In particular, the semigroup algebrasK[T ] are Noetherian maximal orders and domains that satisfy a polyno-mial identity. Moreover, these algebras are intimately connected with settheoretic solutions of the quantum Yang-Baxter equation and Bieberbachgroups. Such algebras have also been studied in [20, 21, 68]. In Section 6.1we will consider equivalent definitions of monoids of I-type and we will il-lustrate, with the help of Theorem 4.1.3, that semigroup algebras of I-typeare indeed Noetherian PI maximal orders and domains. For a survey aboutmonoids of I-type we refer the reader to [27, 43, 45].

In Section 6.2, we use one of the equivalent definitions of a monoid ofI-type to constuct a much wider class of semigroups S, called monoids ofIG-type, and show (again with the help of Theorem 4.1.3) that often theiralgebras K[S] are Noetherian maximal orders that satisfy a polynomialidentity (Theorem 6.2.13). However, these semigroup algebras are not al-ways domains, but we are able to characterize when they are domains, i.e.we describe when the group of quotients of a monoid of IG-type is torsionfree (Theorem 6.2.7). Finally, in the last section we introduce other exam-ples of maximal order monoids that are not necessarily of I- or IG-type.The results in Section 6.2 are joint work with Eric Jespers and have beenpublished in [29].

105

6.1. Monoids of I-type 106

6.1 Monoids of I-type

We first recall the definition of a monoid of I-type. By FaMn we denoteagain the free abelian monoid of rank n with basis {u1, . . . , un}.

Definition 6.1.1. A monoid S, generated by a set X = {x1, . . . , xn}, issaid to be of left I-type if there exists a bijection (called a left I-structure)v : FaMn → S such that

v(1) = 1

and{v(u1a), . . . , v(una)} = {x1v(a), . . . , xnv(a)},

for all a ∈ FaMn. Similarly one defines monoids of right I-type.

In [27] it was shown that a monoid S of left I-type has a presentation

S = 〈x1, . . . , xn | R〉, where R is a set of(n2

)defining relations of the

type xixj = xkxl, so that every word xixj with 1 ≤ i, j ≤ n appears atmost once in one of the relations. Hence, one obtains an associated bijectivemap r : X × X → X × X, defined by r(xi, xj) = (xk, xl) if xixj = xkxl

is a defining relation for S, otherwise one defines r(xi, xj) = (xi, xj). Forevery x ∈ X, denote by fx : X → X and by gx : X → X the mappingsdefined by fx(xi) = p1(r(x, xi)) and gx(xi) = p2(r(xi, x)), where p1 and p2

denote the projections onto the first and second component respectively.So, r(xi, xj) = (fxi(xj), gxj (xi)). One says that r (or simply S) is leftnon-degenerate if each gx is bijective. In case each fx is bijective then r(or S) is said to be right non- degenerate. Also, one says that r is a settheoretic solution of the Yang-Baxter equation if r1r2r1 = r2r1r2, whereri : Xm → Xm is defined as idXi−1 × r × idXm−i−1 and idXj denotes theidentity map on the Cartesian product Xj .

In [27] the equivalence of the first two statements of the following the-orem has been proved. The equivalence with the third statement has beenproved in [43].

Theorem 6.1.1. The following conditions are equivalent for a monoid S.

1. S is a monoid of left I-type.

2. S is finitely generated, say by x1, . . . , xn, and is defined by(n2

)homogeneous relations of the form xixj = xkxl so that every wordxixj with 1 ≤ i, j ≤ n appears at most once in one of the relations


6.1. Monoids of I-type 107

and the associated bijective map r is a solution of the Yang-Baxterequation and is left non-degenerate.

3. S is a submonoid of a semi-direct product of a free abelian monoidFaMn of rank n and a symmetric group of degree n, so that the pro-jection onto the first component is bijective. That is, S = {(a, φ(a)) |a ∈ FaMn} where φ is a mapping from FaMn to Symn so that

φ(a)φ(b) = φ(aφ(a)(b)), (6.1)

or equivalently

φ(ac) = φ(a)φ(φ(a)−1(c)), (6.2)

for all a, b, c ∈ FaMn.

It follows that a monoid is of left I-type if and only if it is of right I-type.Such monoids are simply called monoids of I-type.

Note that the above mentioned semi-direct product FaMn o Symn isdefined via the natural action of Symn on a chosen basis {u1, . . . , un} of thefree abelian monoid FaMn, that is, φ(a)(ui) = uφ(a)(i). Let Fan denote thefree abelian group with the same basis. Then, the monoid S has a group ofquotients SS−1 contained in Fan o Symn and SS−1 = {(a, φ(a)) | a ∈ Fan},where φ : Fan → Symn is a mapping that extends the map FaMn → Symn

and it also satisfies (6.1). In [43] such groups are called groups of I-type.In [27] and [43, 45] it is shown that SS−1 is a solvable Bieberbach group,that is, SS−1 is a finitely generated solvable abelian-by-finite torsion freegroup ([45, Corollary 8.2.7]). These groups also have been investigated byEtingof, Guralnick, Schedler and Soloviev in [20, 21], where they are calledstructural groups.

Gateva-Ivanova and Van den Bergh [27], proved that the semigroupalgebra of such a monoid shares a lot of properties with commutative poly-nomial algebras in finitely many variables. In particular, it is a Noetheriandomain that satisfies a polynomial identity and it is a maximal order.

The same result is obtained by Jespers and Okninski in [43, 45]. Theo-rem 3.1.6 turned out to be the key for proving this. Recall that in case SS−1

is torsion free, Theorem 3.1.6 is equivalent to Theorem 4.1.3 and K[S] is adomain if and only if K[SS−1] is a domain (Theorem 2.3.4). Furthermore,the fact that G = SS−1 is torsion free implies that ∆+(G) = {1}, G isdihedral free and K[G] is a domain (Theorem 2.3.3). We now show thatsemigroup algebras of monoids of I-type satisfy the remaining conditionslisted in Theorem 4.1.3.


6.2. Monoids of IG-type 108

Theorem 6.1.2. Let S be a monoid of I-type in its torsion free finitelygenerated abelian-by-finite group. Then, for every minimal prime ideal P ofS, the intersection P ∩A is SS−1- invariant, where A is an abelian normalsubgroup of SS−1. Furthermore, S is a maximal order that satisfies theascending chain condition on right ideals.

Proof. Theorem 8.5.1 in [45] yields that the semigroup algebra K[S] ofa monoid of I-type is Noetherian and hence, the monoid S satisfies theascending chain condition on right ideals. On the other hand, Lemma 8.5.5in [45] shows that S is a maximal order. From [45, Theorem 8.5.2], weknow that the minimal primes of S are of the form Sfi where fi is a normalelement of S and fn!

i is central. This shows that the intersection P ∩ Ais SS−1-invariant, where A = gr((an!, 1) | a ∈ FaMn), an abelian normalsubgroup of finite index in SS−1. This ends the proof.

6.2 Monoids of IG-type

In this section we introduce a larger class of monoids of interest (monoidsof IG-type). The results have appeared in [29].

Let G be a group and A an abelian monoid. Recall that G is said to acton A if there exists a monoid morphism ϕ : G → Aut(A). The associatedsemi-direct product Aoϕ G we often simply denote by AoG.

Definition 6.2.1. Suppose G is a finite group acting on a cancellativeabelian monoid A. A submonoid S of AoG so that the natural projectionon the first component is bijective is said to be a monoid of IG-type. Thus,

S = {(a, φ(a)) | a ∈ A},

with φ : A→ G a mapping satisfying (6.1) (we denote the action of g ∈ Gon a ∈ A as g(a)).

Note that for every a ∈ A,

(a, φ(a))|G| = (aφ(a) · · ·φ(a)|G|−1(a), φ(a)|G|) = (aφ(a) · · ·φ(a)|G|−1(a), 1).

It follows that b = aφ(a) · · ·φ(a)|G|−1(a) ∈ A is such that φ(b) = 1 and,for every a−1b1 ∈ AA−1 (the group of quotients of A), we have a−1b1 =b−1(φ(a) · · ·φ(a)|G|−1(a)b1). So, any element of AA−1 can be written asa−1b with a, b ∈ A and φ(a) = 1. Furthermore, if a−1

1 b1 = a−12 b2, with

ai, bi ∈ A and φ(ai) = 1, then by equation (6.1), it is easily verified that



φ(b1) = φ(b2). We hence can extend the action of G onto A to an action ofG onto AA−1 and thus obtain a mapping φ : AA−1 → G so that

φ(a−1b) = φ(b),

for every a, b ∈ A with φ(a) = 1. This mapping again satisfies (6.1).Hence S is a submonoid of the group AA−1 o G. Since this group is

abelian-by-finite and S is cancellative, Lemma 2.1.8 yields that S has agroup of quotients SS−1 ⊆ AA−1 o G. Furthermore, because of Theo-rem 2.2.5, the algebra K[S] satisfies a polynomial identity, and, by Lemma3.2.1, SS−1 = SZ(S)−1, with Z(S) the centre of S.

We claim that the natural projection of SS−1 → AA−1 is a one-to-onemapping. Indeed, if a−1

1 b1 = a−12 b2 (with ai, bi ∈ A,φ(ai) = 1) then, by

(6.1), φ(b1) = φ(b2). So,

(a−11 b1, φ(a−1

1 b1)) = (a−11 b1, φ(b1)) = (a−1

2 b2, φ(b2)) = (a−12 b2, φ(a−1

2 b2)).

This proves the injectiveness. The surjectiveness follows from the fact that

(a−11 b1, φ(a−1

1 b1)) = (a−11 b1, φ(b1))

= (a1, 1)−1(b1, φ(b1))= (a1, φ(a1))−1(b1, φ(b1)) ∈ SS−1,

for ai, bi ∈ A with φ(ai) = 1. Groups of the type SS−1 we call groups ofIG-type. So we have shown the following.

Corollary 6.2.2. A group H is of IG-type if and only if H is a subgroupof a semi-direct product AoG of a finite group G with an abelian group Aso that

H = {(a, φ(a)) | a ∈ A}

andφ(aφ(a)(b)) = φ(a)φ(b),

for all a, b ∈ A. Of course, such a group is abelian-by-finite.

Note that if S = {(a, φ(a) | a ∈ A} ⊆ A o G is a monoid of IG-type,with A an abelian monoid and G = {φ(a) | a ∈ A} a finite group then∏

φ(a)∈G φ(a)(b) is an invariant element of A, for every b ∈ A. It followsthat every element of SS−1 can be written as (z, 1)−1(a, φ(a)) with z, a ∈ Aand z an invariant element in A. So (z, 1) is a central element of S.



Lemma 6.2.3. Let A = 〈u1, . . . , un〉 be a finitely generated abelian can-cellativemonoid, G a finite group acting on A. Let S = {(a, φ(a)) | a ∈A} ⊆ A oG be a monoid of IG-type. Put B = {φ(a)(ui) | a ∈ A, 1 ≤ i ≤n}. Then the following conditions hold:

1. G acts on the set B, that is, φ(a)(B) = B, for all a ∈ A.

2. S = 〈(b, φ(b)) | b ∈ B〉.

3. for some divisor k of |G|, the subgroup gr{(bk, 1) | b ∈ B} is normaland of finite index in SS−1.

4. S =⋃

f∈F 〈(bk, 1) | b ∈ B〉(f, φ(f)) and (f, φ(f))〈(bk, 1) | b ∈ B〉 =〈(bk, 1) | b ∈ B〉(f, φ(f)), for some finite subset F of A.

Proof. The first and second part follow at once from the equalities (6.1)and (6.2). Put N = {a ∈ AA−1 | φ(a) = 1}, the kernel of the naturalhomomorphism SS−1 → G. So, N is an abelian subgroup of finite index kin AA−1, with k a divisor of |G|. It follows that φ(ak) = 1, for any a ∈ Aand that the abelian monoid C = 〈(bk, 1) | b ∈ B〉 is contained in S and itsgroup of quotients CC−1 = gr{(bk, 1) | b ∈ B} is normal and of finite indexin SS−1. This proves the third part. Part four is now also clear.

If, in the previous lemma, U(A) = {1}, then one can take {u1, . . . , un}to be the set of indecomposable elements, that is, the set consisting ofthose elements f ∈ A so that Af is a maximal principal ideal. Clearly anyautomorphism of A permutes the indecomposable elements and it followsthat S = 〈(u1, φ(u1)), . . . , (un, φ(un))〉.

We describe now when the semigroup algebra K[S] of a monoid of IG-type is Noetherian. This result follows immediately from Lemma 3.1.5.

Proposition 6.2.4. Let A be an abelian cancellative monoid and G a finitegroup acting on A. Let S = {(a, φ(a)) | a ∈ A} ⊆ A o G be a monoid ofIG-type. Then, the semigroup algebra K[S] is Noetherian if and only ifthe abelian monoid A is finitely generated, or, equivalently, S is finitelygenerated.

Let us now investigate periodic elements of a monoid S = {(a, φ(a)) |a ∈ A} of IG-type. We will assume that the action of G = {φ(a) | a ∈ A}on A is faithful. Because of the latter condition we may consider G asa subgroup of the automorphism group of A. If A is finitely generated,we know that A has only finitely many minimal prime ideals and everyprime ideal is a union of minimal prime ideals. Also remark that, if A is an



abelian cancellative maximal order and U(A) = {1}, then AA−1 is torsionfree. Indeed, if a ∈ AA−1 is such that an ∈ A, for some natural number n,then A being a maximal order implies that a ∈ A. Hence it follows that(AA−1)+ ⊆ U(A) = {1}, where (AA−1)+ is the torsion part of AA−1. Soindeed, AA−1 is torsion free.

Proposition 6.2.5. Let A be an abelian cancellative monoid. Assume thatA is a finitely generated maximal order and U(A) = {1}. If S = {(a, φ(a)) |a ∈ A} ⊆ AoG is a monoid of IG-type and the action of G = {φ(a) | a ∈ A}is faithful, then

S ∼= (D(A)+ oG) ∩ SS−1,

where D(A) is the divisor class group of A, D(A)+ its positive cone and Ga subgroup of the permutation group of the minimal primes of A.

Proof. As A is an abelian Krull order, we know that D(A) is a free abeliangroup with the set X1(A) consisting of the minimal primes of A as a freegenerating set. Of course, for each a ∈ A, φ(a) induces an automorphismon X1(A), and thus also on D(A). We denote this again by φ(a). It followsthat, if I is an ideal of S, then φ(a)(I∗) = (φ(a)(I))∗. We thus obtain amorphism G → Sym(X1(A)). This mapping is injective. Indeed, supposeφ(a) is the identity map on X1(A), with a ∈ A. For c ∈ A the ideal Acis divisorial. Hence Aφ(a)(c) = φ(a)(Ac) = Ac. Since, by assumptionU(A) = {1}, it follows that φ(a)(c) = c. Hence it follows that φ(a) = 1.Because, also by assumption, the action of G on A is faithful, it followsthat φ(a) is the identity map on G, as desired.

Again, because U(A) = {1}, we get a monoid morphism

S → D(A)+ oG : (a, φ(a)) 7→ (aA, φ(a)).

So, identifying S with its image in D(A)+ o G (and also SS−1 with itsimage in {(a−1bA, φ(a−1b)) | a, b ∈ A} ⊆ D(A) oG), we get that

S ⊆ (D(A)+ oG) ∩ SS−1.

Conversely, if b ∈ AA−1 and (bA, φ(b)) ∈ D(A)+ o G, then bA ⊆ A andthus b ∈ A. Hence (bA, φ(b)) ∈ S.

Note that this characterization is a non-commutative version of theresult of Chouinard (Theorem 5.0.1.) that describes commutative cancella-tive semigroups that are Noetherian maximal orders or more generally Krullorders.

To investigate the torsion freeness, we need the following theorem, whichis proved by Karel Dekimpe ([18]).



Theorem 6.2.6. Let H be a group of affine transformations such thatH ∩ Rn (the subgroup of pure translations) is of finite index in H. Thenthe following properties are equivalent.

1. H is torsion free.

2. The action of H on Rn is fixed-point free, that is, if g ·a = a for somea ∈ Rn and g ∈ H, then g = 1.

Note that, if SS−1 is torsion free, then so is necessarily AA−1. Indeed,If am = 1 in AA−1, then, by Lemma 6.2.3,

(a, φ(a))km = (aφ(a)(a)φ(a)2(a)...φ(a)k−1(a), 1)m = 1 ∈ SS−1,

for some divisor k of |G|.Assume now that S = {(a, φ(a)) | a ∈ A} is a monoid of IG-type

with faithful action of G = {φ(a) | a ∈ A} on A. Suppose that AA−1 isa torsion free finitely generated abelian group. So, SS−1 ⊆ Zk o G andG ⊆ Aut(Zk) ∼= GLk(Z). Hence, every element of G can be seen as a k×k-matrix with values in Z and the action of SS−1 on Zk can be extended toRk and can be written as:

(a, φ(a)) · b = φ(a)b+ a,

where a ∈ Zk, φ(a) ∈ GLk(Z), b ∈ Zk (or Rk) and φ(a)b is given by theclassical matrix multiplication. For convenience sake, we use the additivenotation on Zk and Rk (instead of the multiplicative on AA−1). So, SS−1 ⊆Rk o GLk(R). Thus, SS−1 is a group of affine transformations and everyelement of SS−1 is of the form (a, φ(a)), where a is the translation partand φ(a) the linear part. Clearly, (a,A)(b, B) = (a + Ab,AB). As G isa finite group, we have that the subgroup of pure translations is of finiteindex in SS−1. Then, as an immediate consequence of Theorem 6.2.6,we get that the quotient group SS−1 ⊆ AA−1 o G is torsion free if andonly if the action of SS−1 on Rn is fixed-point free. If also U(A) = {1}and A is a maximal order in its free abelian group of quotients then, byProposition 6.2.5, we can extend the action of SS−1 to an action of thesemi-direct product D(A) o G and G acts as the symmetric group on theset X1(A) = {P1, ..., Pl}.

Theorem 6.2.7. Let A be an abelian cancellative monoid. Assume A is afinitely generated maximal order with U(A) = {1}. If S = {(a, φ(a)) | a ∈A} ⊆ AoG is a monoid of IG-type and the action of G = {φ(a) | a ∈ A}on A is faithful, then an element (a, φ(a)) of SS−1 is periodic if and onlyif there exists a divisorial ideal I of A such that aφ(a)(I) = I.



Proof. Suppose (a, φ(a)) is a periodic element. So, because of Proposition6.2.5 and Theorem 6.2.6, (aA, φ(a)) ∈ D(A) o Syml has a fixed point b inRl. Write aA = Pα1

1 ∗ ... ∗ Pαll , with αi ∈ Zl and X1(A) = {P1, ..., Pl}. So

φ(a) · b+α = b, where α = (α1, ..., αl). Since φ(a) acts as a permutation onthe components of b ∈ Rl, it is not so difficult to see that φ(a)·bbc+α = bbc,where bbc is the integral part of b. Hence we have a fixed point in Zl. Thismeans that aφ(a)(I) = I, where I = P β1

1 ∗ ... ∗ P βll , with (β1, ..., βl) = bbc.

Conversely, suppose there exists a divisorial ideal I of A such thataφ(a)(I) = I. It follows that

a φ(a)(a) φ(a)2(a) · · · φ(a)n(a) φ(a)n+1(I) = I.

So if φ(a)n+1 = 1, then we obtain that a φ(a)(a) · · · φ(a)n(a)A = A.Again because U(A) = 1, it follows that a φ(a)(a) · · · φ(a)n(a) = 1.Consequently, (a, φ(a))n+1 = 1, as desired.

Remark that although the group of quotients SS−1 is not always torsionfree (so the semigroup algebra K[S] is not always a domain), the semigroupalgebra K[S] of a monoid of IG-type is always prime, provided AA−1 istorsion free (in particular, if A is an abelian cancellative maximal orderand U(A) = {1}).

Theorem 6.2.8. Let A be an abelian cancellative monoid such that AA−1

is torsion free. Then the semigroup algebra K[S] of the monoid of IG-typeS = {(a, φ(a)) | a ∈ A} is prime.

Proof. From Theorem 2.3.3 and Theorem 2.3.4, it follows that we haveto show that a non-trivial periodic element in SS−1 has infinitely manyconjugates in SS−1. So, suppose (a, φ(a)) is a non-trivial periodic elementin SS−1. Because of the assumption that AA−1 is torsion free, we get thatφ(a) 6= 1. Hence, there exists an element b ∈ AA−1 such that φ(a)(b) 6=b. Again, by the torsion freeness, it follows that φ(a)(bm) 6= bm, for allm ∈ N0. Hence, replacing if necessary, b by a power of b, we can assumethat φ(b) = 1. Then, for every n ∈ N, the element (bnaφ(a)(b−n), φ(a)) isconjugated with (a, φ(a)). Clearly, the set bnφ(a)(b−n) = (bφ(a)(b−1))n isinfinite, as φ(a)(b) 6= b and AA−1 is torsion free.

We illustrate Theorem 6.2.8 with two examples of monoids of IG-type.The first example is torsion free, while the second one is not. Both examplesare build on the abelian Anderson monoid, which is discussed in Chapter 4.



Example 6.2.9. Let A = 〈u1, u2, u3, u4 | u1u2 = u3u4〉 and let | | : A→ Zdenote the degree function on A defined by |ui| = 1. Put

σ =

0 1 11 0 10 0 −1

∈ Gl3(Z).

The natural action of Z2 = 〈σ〉 on Z3 = AA−1 = gr(u1, u2, u3) defines asemi-direct product Ao Z2. Then

S = {(a, φ(a)) | a ∈ A, φ(a) = 1 if |a| ∈ 2Z, φ(a) = σ if |a| ∈ 2Z + 1}

is a monoid of IG-type and its group of quotients SS−1 is

{(a, φ(a)) | a ∈ Z3, φ(a) = 1 if |a| ∈ 2Z, φ(a) = σ if = |a| ∈ 2Z + 1}

is torsion free.

Proof. That A is a maximal order in its torsion free group of quotientsfollows from Theorem 5.1.2. Clearly U(A) = {1} and A has four minimalprimes: Q1 = (u1, u3), Q2 = (u1, u4), Q3 = (u2, u3) and Q4 = (u2, u4). So,these minimal primes generate the free abelian group D(A) ∼= Z4. Becauseof Theorem 6.2.7, to prove that SS−1 is torsion free, we need to show thatif (a, φ(a)) ∈ SS−1 is such that aφ(a)(I) = I for some divisorial ideal Iof A then a = 1. Clearly, if |a| ∈ 2Z then φ(a) = {1}, and thus aI = Iimplies a = 1. So, suppose a has odd degree. Then a = ua1

1 ua22 u

a33 or

a = ua11 u

a22 u

a44 . We deal with the former case (the other case is dealt with

similarly). It is readily verified that Au1 = Q1 ∗ Q2, Au2 = Q3 ∗ Q4 andAu3 = Q1 ∗Q3. Write

I = Qγ11 ∗Qγ2

2 ∗Qγ33 ∗Qγ4

4 ,

with each γi ∈ Z. Because σ interchanges Q1 with Q4 and Q2 with Q3, theequality Aa ∗ φ(a)(I) = (Aaφ(a)(I))∗ = I becomes

Qa1+a31 ∗Qa1

2 ∗Qa2+a33 ∗Qa2

4 ∗Qγ14 ∗Qγ2

3 ∗Qγ32 ∗Qγ4

1 = Qγ11 ∗Qγ2

2 ∗Qγ33 ∗Qγ4

4 .

It follows that a1 + a2 + a3 = 0, in contradiction with the fact that a is ofodd degree.

Example 6.2.10. Let A = 〈u1, u2, u3, u4 | u1u2 = u3u4〉 and let D8 =〈a, b | a4 = 1, b2 = 1, a3b = ba〉, with a = (1324) and b = (12), the dihedral



group of order 8. So D8 acts naturally on A. In the semidirect productAoD8 consider the elements xi = (ui, σi), where

σ1 = (1324), σ2 = (12), σ3 = (1423) and σ4 = (34)

and let S = 〈x1, x2, x3, x4〉. Then S is a monoid of IG-type. Furthermore,SS−1 has non-trivial periodic elements.

Proof. It is easily verified that S = {(a, φ(a)) | a ∈ A} and thus S is amonoid of IG-type, with G = {φ(a) | a ∈ A} = D8. So, SS−1 ⊆ AA−1oD8.Clearly,

(u3, σ3)(σ−11 (u−1

1 ), σ−11 ) = (u3u

−12 , (12)(34)),

and, as u4 = u1u2u−13 in AA−1, we have that (u3u

−12 , (12)(34))2 = 1. So

SS−1 has non-trivial periodic elements, although it follows from Theo-rem 6.2.8 and the fact that AA−1 is torsion free, that K[S] is prime.

Remark that Theorem 5.1.2 and Theorem 5.2.2 give us plenty exam-ples of finitely generated abelian maximal orders A on which we can buildexamples of monoids of IG-type.

In order to prove a characterization of semigroup algebras K[S] of IG-type that are a maximal order, we describe now the prime ideals of S.Corollary 3.2.5 turns out to be the tool for doing this. For an ideal I of A,we put (I, φ(I)) = {(a, φ(a)) | a ∈ I}. Note that this is a right ideal of S.

Theorem 6.2.11. Let S = {(a, φ(a)) | a ∈ A} ⊆ A o G be a monoidof IG-type, with A a finitely generated abelian cancellative monoid suchthat AA−1 torsion free. The prime ideals P of S of height m are the sets(Q1 ∩ ... ∩Qn, φ(Q1 ∩ ... ∩Qn)) so that

1. each Qi is a prime ideal of A of height m,

2. aφ(a)(Q1 ∩ ... ∩Qn) ⊆ Q1 ∩ ... ∩Qn, for every a ∈ A(that is, (Q1 ∩ ... ∩Qn, φ(Q1 ∩ ... ∩Qn)) is an ideal of S),

3. condition (2) is not satisfied for an intersection over a proper subsetof {Q1, ..., Qn} (that is, (Q1 ∩ ...∩Qn, φ(Q1 ∩ ...∩Qn)) is a maximalset satisfying conditions (1) and (2)).

Proof. Let P be a prime ideal of S and let Ak = {ak | a ∈ A}, where k isa divisor of the order of the group G such that φ(ak) = 1, for every a ∈ A(see Lemma 6.2.3). We identify the group AkA−k with its natural image



in SS−1 and it follows that this group is of finite index in SS−1. FromCorollary 3.2.5, we get that

P ∩AkA−k = P ∩ S ∩AkA−k = Q(k)1 ∩ · · · ∩Q(k)

n ,

where Q(k)i are prime ideals of T = S ∩AkA−k. Since S ∩AkA−k is SS−1-

invariant, it follows from Corollary 3.2.5 that the primesQ(k)i are conjugated

and it is easily verified that they are of the same height as P .Remark that we have a bijection between the prime ideals of A and Ak.

Indeed, let Q be a prime ideal of A, then Qk = {qk | q ∈ Q} ⊆ Q ∩Ak andQk is a prime ideal of Ak. Furthermore Q ∩ Ak is a nil ideal modulo Qk.Since Ak is commutative, it follows that Q∩Ak ⊆ Qk. Hence Q∩Ak = Qk.So we have a bijection and corresponding primes have the same height.Clearly, Ak ⊆ T ⊆ A. It is then easily verified that the following mapyields a bijection between the prime ideals of A and those of T :

Q 7→ Q ∩ T.

Therefore Q(k)i = Qi ∩ T , for a prime ideal Qi in A of the same height as

P . If (a, φ(a)) ∈ P then

(a, φ(a))(φ(a)−1(a), φ(φ(a)−1(a)))... = (ak, 1) ∈ P ∩ T.

Hence ak ∈ Q(k)1 ∩ · · · ∩Q(k)

n , and thus a ∈ Q1 ∩ · · · ∩Qn. Therefore,

P ⊆ (Q1 ∩ · · · ∩Qn, φ(Q1 ∩ · · · ∩Qn)).

Conversely, if (b, φ(b)) ∈ (Q1 ∩ · · · ∩Qn, φ(Q1 ∩ · · · ∩Qn)), then

((b, φ(b))k)k ∈ (Q1 ∩ · · · ∩Qn, φ(Q1 ∩ · · · ∩Qn)) ∩ T ⊆ ∩ni=1Q

(k)i ⊆ P.

So (Q1∩· · ·∩Qn, φ(Q1∩· · ·∩Qn)) is a right ideal of S that is nil modulo Pand thus we get from Theorem 2.1.3 that (Q1∩· · ·∩Qn, φ(Q1∩· · ·∩Qn)) ⊆P . Hence P = (Q1 ∩ · · · ∩Qn, φ(Q1 ∩ · · · ∩Qn)).

Since P is a left ideal, it is clear that aφ(a)(Q1∩· · ·∩Qn) ⊆ Q1∩· · ·∩Qn.Now we show that if P1 = (Q1 ∩ · · · ∩ Qn, φ(Q1 ∩ · · · ∩ Qn)) and P2 =(Q′1 ∩ · · · ∩ Q′m, φ(Q′1 ∩ · · · ∩ Q′m)) are different prime ideals (of the sameheight) of S then {Q1, · · · , Qn} ∩ {Q′1, · · · , Q′m} = ∅. Indeed, suppose thecontrary, then, without loss of generality, we may assume that Q1 = Q′1.As P1 6= P2, and because they are of the same height, we thus get that say



n > 1 and m > 1. Clearly (Q2 ∩ · · · ∩Qn, φ(Q2 ∩ · · · ∩Qn)) is a right idealof S and

(Q2 ∩ · · · ∩Qn, φ(Q2 ∩ · · · ∩Qn))P2

⊆ {(aφ(a)(Q′1 ∩ · · · ∩Q′m), φ(aφ(a)(Q′1 ∩ · · · ∩Q′m))) | a ∈ Q2 ∩ · · · ∩Qn}⊆ ((Q′1 ∩ · · · ∩Q′m) ∩ (Q2 ∩ · · · ∩Qn), φ(Q′1 ∩ · · · ∩Q′m ∩Q2 ∩ · · · ∩Qn))⊆ P1

Because ht(Q1) = ht(Q2) = · · · = ht(Qn) and the primes Q1, . . . , Qn aredistinct, we get that Q2 ∩ · · · ∩ Qn * Q1 ∩ · · · ∩ Qn. As P1 is prime wethus obtain that P2 ⊆ P1. Since they are of the same height, it follows thatP1 = P2, a contradiction. The above claim of course implies the minimalityas stated in the Theorem.

To end the proof, we need to show that the ideals (Q1∩· · ·∩Qn, φ(Q1∩· · · ∩ Qn)) with the listed properties are prime ideals of S. We know thatQ

(k)1 = Q1 ∩ T is a prime ideal of T of the same height as Q1. Then, using

Corollary 3.2.5 and the first part of the proof, we know that there existsa prime ideal P1 of S that lies over Q(k)

1 so that P1 = (Q1 ∩ Q′2 ∩ · · · ∩Q′m, φ(Q1 ∩Q′2 ∩ ... ∩Q′m)) with Q′i primes in A of the same height as Q1.We can now do the same for Q2, . . . , Qn. Hence, we get primes P2, . . . , Pn

of S so that P2 = (Q2 ∩ J2, φ(Q2 ∩ J2)), . . . , Pn = (Qn ∩ Jn, φ(Qn ∩ Jn)),with each Ji an intersection of primes of A that are of the same height asQi and we agree that Ji = A if Pi = (Qi, φ(Qi)). Furthermore, because ofthe assumptions we get that

(J2, φ(J2))(Q1 ∩ · · · ∩Qn, φ(Q1 ∩ · · · ∩Qn))⊆ (J2 ∩Q1 ∩ · · · ∩Qn, φ(J2 ∩Q1 ∩ · · · ∩Qn)) ⊆ P2.

Since (J2, φ(J2)) * P2 this yields that (Q1 ∩ · · · ∩Qn, φ(Q1 ∩ · · · ∩Qn)) ⊆P2 = (Q2 ∩ J2, φ(Q2 ∩ J2)). Since {Q1, . . . , Qn} satisfies the minimalitycondition as stated in the Theorem, we obtain that Q1∩· · ·∩Qn ⊆ Q2∩J2.Then P1 ⊆ P2 and, as they have the same height, P1 = P2. Thus P1 =P2 = · · · = Pn = (Q1 ∩ · · · ∩Qn, φ(Q1 ∩ · · · ∩Qn)) and thus this is a primeideal of S.

Corollary 6.2.12. If L = {Q1, ..., Qn} is a full G-orbit of primes of thesame height in A, then there exists a partition {X1, ..., Xn} of L, so that

(∩Q∈XiQ,φ(∩Q∈XiQ)) = Pi

are prime ideals of S.



As a consequence of Theorem 4.1.3, we now prove the main result ofthis chapter. It provides a characterization of semigroup algebras K[S] ofmonoids of IG-type that are a maximal order. It turns out that the struc-ture of the monoid A is crucial in order to obtain K[S] a prime Noethe-rian maximal order and thus we even become a stronger result than Theo-rem 4.1.3.

Theorem 6.2.13. Let S = {(a, φ(a)) | a ∈ A} ⊆ AoG be a monoid of IG-type, with A a finitely generated abelian cancellative monoid such that AA−1

torsion free. Then, the Noetherian prime ring K[S] is a maximal order ifand only if the abelian monoid A is a maximal order and the minimalprimes of S are of the form

P = (Q1 ∩ · · · ∩Qn, φ(Q1 ∩ · · · ∩Qn)),

where {Q1, · · · , Qn} = {φ(a)(Q1) | a ∈ A} ⊆ X1(A).

Proof. That the semigroup algebra K[S] is Noetherian and prime followsfrom Theorem 6.2.4 and Theorem 6.2.8. Because of Theorem 4.1.3, in orderto prove the sufficiency, we only need to verify that the conditions implythat S is a maximal order. Suppose therefore that the minimal primes ofS are of the from (Q1 ∩ .... ∩Qn, φ(Q1 ∩ .... ∩Qn)), where {Q1, ...., Qn} ={φ(a)(Q1) | a ∈ A} ⊆ X1(A) and that A is a maximal order. We show nowthat S is a maximal order. Because of Lemma 2.5.10, in order to prove thisproperty, it is sufficient to show that (P :l P ) = (P :r P ) = S for everyprime ideal P of S. From Theorem 6.2.11 we know that P = (Q,φ(Q)) withQ an intersection of prime ideals in A of the same height, say n. Assume(x, φ(x)) ∈ (P :l P ). Then xφ(x)(Q) ⊆ Q. If n 6= 0 (so Q is an intersectionof primes that are not minimal) then the divisorial closure of both Q andφ(x)(Q) equals A. As x(φ(x)(Q))∗ ⊆ Q∗ we thus get that x ∈ A andthus (x, φ(x)) ∈ S. If n = 0, then, by assumption, Q is G-invariant andthus we get that xQ ⊆ Q. Since A is a maximal order, this yields thatx ∈ A and again (x, φ(x)) ∈ S. So (P :l P ) = S. On the other hand,suppose (Q,φ(Q))(x, φ(x)) ⊆ (Q,φ(Q)). Then A(Q ∩ Ak)x ⊆ Q. If n 6= 0then Q ∩ Ak is not contained in a minimal prime ideal of A and thus thedivisorial closure of both A(Q∩Ak) and Q is A. Since (A(Q∩Ak))∗x ⊆ Q∗

we get that (x, φ(x)) ∈ S. So it remains to show that (P :r P ) = S fora minimal prime ideal P . Hence, by assumption P = (Q,φ(Q)) with Q aG-invariant ideal. More generally, we prove that (I :r I) = S for any idealI = (M,φ(M)) of S with M a G-invariant ideal of A.

We now prove this by contradiction. So suppose that I is such an idealof S with Ig ⊆ I for some g ∈ SS−1 \ S.



Now, as g ∈ SS−1, we know that g = (a, φ(a))(z, 1)−1 with z an invari-ant element of A, and thus (z, 1) central in S. As A is a maximal order, wehave that the minimal primes of A freely generate the abelian group D(A).So, in the divisor group D(A), we can write Az as a product of minimalprimes. Because Az is invariant, the minimal primes in a G-orbit have thesame exponent. Hence,

Az = (Jn11 )∗ ∗ · · · ∗

(Jnl

l

)∗,

where each Ji is an intersection of all minimal primes of A in a G-orbit.So, because of the assumption and Theorem 6.2.11, each (Ji, φ(Ji)) is aminimal prime of S. Of course also Aa is a divisorial product of minimalprimes of A. If necessary, cancelling some common factors of Aa and Az,we may assume that Aaz−1 = K ∗L−1 6⊆ A, and thus KL−1 6⊆ A with L =(Jn1

1 )∗ ∗ · · · ∗(Jnl

l

)∗, L−1 = (A : L) and K is not contained in Ji, for every iwith 1 ≤ i ≤ l. Note that, also, L−1 is G-invariant and thus (L−1, φ(L−1))is a fractional ideal of S. Of course, I(K,φ(K))(L−1, φ(L−1) ⊆ I. BecauseS satisfies the ascending chain condition on ideals, we can choose I maximalwith respect to the property that such K and L exist with KL−1 6⊆ A.

Clearly we obtain that

I(K,φ(K))(L−1, φ(L−1))(L, φ(L)) ⊆ I(L, φ(L)) ⊆ S(L, φ(L))⊆ (Jni

i , φ(Jnii ))

⊆ (Ji, φ(Ji)).

Since, (Ji, φ(Ji)) is a prime ideal of S, we get that either (KL−1L, φ(KL−1L))⊆ (Ji, φ(Ji)) or I ⊆ (Ji, φ(Ji)). Because of the above, the former is ex-cluded. Hence I ⊆ (Ji, φ(Ji)). As Ji is G-invariant, we get again that(J−1

i , φ(J−1i )) is a fractional ideal of S that contains S. Therefore, we get

that I ⊆ (J−1i , φ(J−1

i ))I is an ideal of S. Since

(J−1i , φ(J−1

i ))I(K,φ(K))(L−1, φ(L−1)) ⊆ (J−1i , φ(J−1

i ))I,

the maximality condition on I thus implies that

(J−1i , φ(J−1

i ))I = I.

Since M is G-invariant this yields that J−1i M = M and thus J−1

i ∗M∗ =M∗. So J−1

i = A, a contradiction.To prove the necessity of the conditions, assume K[S] is a maximal

order. From Lemma 3.4.1 and Lemma 4.1.1, it follows that A is a maximal



order. On the other hand, let P = (M,φ(M)) be a minimal prime ideal ofS. Theorem 4.1.3 yields that P∩Ak = M∩Ak isG-invariant. Consequently,M is G-invariant and Theorem 6.2.11 yields that M is the intersection ofa full G-orbit of minimal primes. This finishes the proof.

As an illustration of Theorem 6.2.13 we give two examples. One thatis a maximal order and one that is not. These examples yield the answersto the questions posed in Example 3.1.2 and Example 3.1.3.

Example 6.2.14. Let S ⊆ Ao Z2 be the monoid of IG-type considered inExample 6.2.9. The semigroup S is a maximal order and

P1 = (Q1 ∩Q4, φ(Q1 ∩Q4)),

P2 = (Q2 ∩Q3, φ(Q2 ∩Q3))

are its minimal prime ideals. Furthermore, K[S] is a maximal order forany field K.

Proof. Since S is

{(a, φ(a)) | a ∈ A, φ(a) = 1 if |a| ∈ 2Z, φ(a) = (12)(34) if |a| ∈ 2Z + 1},

it is easily verified that S = 〈x1, x2, x3, x4〉, with xi = (ui, φ(ui)) and withthe following relations:

x21 = x2

2 = x23 = x2

4

x1x3 = x4x2, x1x4 = x3x2

x2x3 = x4x1, x2x4 = x3x1.

Indeed, as

x21 = (u1, (12)(34))(u1, (12)(34)) = (u1u2, 1)

= (u2u1, 1) = (u2, (12)(34))(u2, (12)(34)) = x22,

x23 = (u3, (12)(34))(u3, (12)(34)) = (u3u4, 1)

= (u4u3, 1) = (u4, (12)(34))(u4, (12)(34)) = x24

and x21 = (u1u2, 1) = (u3u4, 1) = x2

3, the first equation follows. On theother hand,

x1x3 = (u1, (12)(34))(u3, (12)(34)) = (u1u4, 1)= (u4u1, 1) = (u4, (12)(34))(u2, (12)(34)) = x4x2



and the other relations are completely similar. Hence, the semigroup al-gebra of this monoid of IG-type is the algebra in Example 3.1.1. FromExample 6.2.9 (and Theorem 5.1.2) we know that A is a finitely gener-ated maximal order with minimal primes: Q1 = (u1, u3), Q2 = (u1, u4),Q3 = (u2, u3) and Q4 = (u2, u4). Clearly, aφ(a)(Q1 ∩ Q4) ⊆ (Q1 ∩ Q4)and aφ(a)(Q2 ∩ Q3) ⊆ (Q2 ∩ Q3). Let P1 = (Q1 ∩ Q4, φ(Q1 ∩ Q4)) andP2 = (Q2 ∩ Q3, φ(Q2 ∩ Q3)). Hence, because of Theorem 6.2.11, to provethat P1 and P2 are the only minimal primes of S, it is now sufficient tonote that for every Qi, there exists an a ∈ A such that aφ(a)(Qi) * Qi.Indeed, u2(12)(34)Q1 * Q1, u2(12)(34)Q2 * Q2, u4(12)(34)Q3 * Q3, andu3(12)(34)Q4 * Q4. Thus the result follows from Theorem 6.2.13.

Example 6.2.15. Let A = 〈u1, u2, u3, u4 | u1u2u3 = u24〉 be an abelian

monoid. Then A is a maximal order (in its torsion free group of quotients)with minimal prime ideals

Q1 = (u1, u4) , Q2 = (u2, u4) and Q3 = (u3, u4).

Let S = {(a, φ(a)) | a ∈ A} ⊆ AoZ2, with φ(a) = 1 if a ∈ A has even degreein u4, otherwise, φ(a) = (12) (the transposition interchanging u1 with u2).Then, S = 〈x1, x2, x3, x4〉 is a monoid of IG-type with xi = (ui, φ(ui)) andwith the following relations:

x1x2x3 = x24

x1x2 = x2x1, x1x3 = x3x1, x2x3 = x3x2

x1x4 = x4x2, x2x4 = x4x1, x3x4 = x4x3.

Furthermore, S is not a maximal order and the group of quotients SS−1

is torsion free. Thus, K[S] is not a maximal order for any field K. Theminimal prime ideals of S are Pi = (Qi, φ(Qi)) with 1 ≤ i ≤ 3.

Proof. From Theorem 5.1.2, it follows that A is a maximal order. As Z2 =gr((12)) induces a faithful action on the finitely generated monoid A =〈u1, u2, u3, u4 | u1u2u3 = u2

4〉, we thus get that S ⊆ A o Z2 is a monoid ofIG-type. From

x24 = (u4, (12))(u4, (12)) = (u2

4, 1) = (u1u2u3, 1)= (u1, 1)(u2, 1)(u3, 1) = x1x2x3,

the first equation follows. For the remainder, we only show that x1x4 =x4x2 as the other relations are similar. Indeed,

x1x4 = (u1, 1)(u4, (12)) = (u1u4, (12))= (u4u1, (12)) = (u4, (12))(u2, 1) = x4x2,



the semigroup algebra of this monoid of IG-type is the algebra in Exam-ple 3.1.2. Suppose that there exists a non-trivial periodic element in thegroup of quotients SS−1. Such an element must be of the form

(uα11 uα2

2 uα44 , (12))

with αi ∈ Z and α4 odd. Then, by Theorem 6.2.6 (see also the remarksstated after its proof), (α1, α2, α4)+(12)(α′1, α

′2, α

′4) = (α′1, α

′2, α

′4), for some

α′1, α′2, α

′4 ∈ R. Hence α4 + α′4 = α′4 and thus α4 = 0, a contradiction. So,

SS−1 indeed is torsion free.Let I be the ideal generated by ((u1, 1), (u4, (12))). Then

(u1u3u−14 , (12))I ⊆ I

and thus S is not a maximal order. As an immediate consequence ofTheorem 6.2.11 one sees that P1 = (Q1, φ(Q1)), P2 = (Q2, φ(Q2)) andP3 = (Q3, φ(Q3)) are the minimal prime ideals of S. Clearly Q1 and Q2

are not Z2-invariant.

We finish this section by showing a nice link between monoids of I-typeand monoids of IG-type.

Theorem 6.2.16. A finitely generated monoid S is of IG-type if andonly if there exists a monoid of I-type T = {(x, ψ(x)) | x ∈ FaMm} ⊆FaMm o Symm and a subgroup B of Fam that is ψ-invariant so that S ∼=TB/B.

Proof. Assume S = {(a, φ(a)) | a ∈ A} ⊆ A o G is a finitely generatedmonoid of IG-type, where G is a finite group acting on the finitely generatedabelian monoid A = 〈u1, . . . , un〉. Of course we may assume that G ={φ(a) | a ∈ A}.

Let m = n|G| and let FaMm be the free abelian monoid of rank mwith basis the set M = {vg,i | g ∈ G, 1 ≤ i ≤ n}. Clearly the mappingf : FaMm → A defined by f(vg,i) = gui is a monoid epimorphism. For x ∈FaMm define a mapping ψ(x) : M → M by ψ(x)(vg,i) = vφ(f(x))g,i. Thenψ(x) ∈ Symm, f(ψ(x)(y)) = φ(f(x))(f(y)) and ψ(xψ(x)(y)) = ψ(x)ψ(y),for any x, y ∈ FaMm. So T = {(x, ψ(x)) | x ∈ FaMm} is a monoid ofI-type contained in FaMm o Symm. Furthermore, fe : T → S defined byfe((x, ψ(x))) = (f(x), φ(f(x))) is a monoid epimorphism. Its extension toan epimorphism TT−1 → SS−1 we also denote by fe. Let B = ker(fe).Clearly, if (x, ψ(x)) ∈ B then φ(f(x)) = 1 and thus ψ(x) = 1. Thus,


6.3. Other examples 123

B ⊆ Fam and B is ψ-invariant. So TB = BT is a submonoid of TT−1 andTB/B ∼= S. This proves the necessity of the conditions.

Conversely, assume T = {(x, ψ(x)) | x ∈ FaMm} ⊆ FaMm o Symm is amonoid of I type and B is a subgroup of Fam that is ψ-invariant. Let A =FaMmB/B and let f : FaMm → A be the natural monoid epimorphism.Because of (6.2) we get that ψ(x) = ψ(y) if f(x) = f(y). Hence, for eacha = f(x) the mapping φ(a) : A→ A given by φ(a)(f(y)) = f(ψ(x)(y)) is awell defined bijection of finite order. Furthermore, φ(aφ(a)(b)) = φ(a)φ(b)for all a, b ∈ A. Hence S = {(a, φ(a)) | a ∈ A} is a monoid of IG-typecontained in A o G, where G = {φ(a) | a ∈ A}. The mapping TB → Sdefined by mapping (x, ψ(x)) onto (a, φ(a)) is a monoid epimorphism andit easily follows that this map induces an isomorphism between TB/B andS.

The proposition can be formulated using congruence relations as follows.A finitely generated monoid S is of IG-type if and only if there exists amonoid of I-type T = {(a, ψ(a)) | a ∈ FaMm} ⊆ FaMm o Symm and thereexists a congruence relation ρ on FaMm with

a ρ b implies ψ(a) = ψ(b) and ψ(x)(a) ρ ψ(x)(b), (6.3)

for every a, b, x ∈ FaMm, and so that S ∼= T/ρ where ρ is the congru-ence relation on T defined by (a, ψ(a)) ρ (b, ψ(b)) if and only if a ρ b, fora, b ∈ FaMm. We remark that many monoids of IG-type are not of I-type.Indeed, suppose S = {(a, φ(a)) | a ∈ A} is a monoid of IG-type with A afinitely generated monoid so that U(A) = {1}. Let {u1, . . . , un} be the setof indecomposable elements of A. So A = 〈u1, . . . , un〉. It follows that theelements (ui, φ(ui)) are the unique indecomposable elements of S, that is,they can not be decomposed as a product of two non-invertible elements.So S also has n indecomposables. In particular, the number of indecom-posables in a monoid T of I-type equals the torsion free rank of any abeliansubgroup of finite index in TT−1. So, if the torsion free rank of AA−1 isstrictly smaller than n then S is not of I-type.

6.3 Other examples

In Chapter 5, we classified all finitely generated abelian maximal orders,given by a presentation with 1 or 2 defining monomial relations (Theo-rems 5.1.2, 5.2.2). In order to construct more non-abelian examples, let usconcentrate on these abelian finitely presented maximal order monoids A.



Let G be a finite group that acts on A and consider the full semi-directproduct, AoG. So, in contrary to the IG-type monoids, we do not imposethat the projection on the first component is bijective. These classes ofsemigroups will yield new examples of maximal orders.

Lemma 6.3.1. Let A be a finitely presented abelian cancellative monoidand let G be a finite group that acts on A. Let S = AoG be the correspond-ing semidirect product. Then S satisfies the ascending chain condition onone-sided ideals, and, if A is a maximal order in its torsion free group ofquotients, then S is a maximal order in an abelian-by-finite group.

Proof. That S satisfies the ascending chain condition on one-sided ideals,follows from Lemma 3.1.5 and the fact that A is finitely generated. Assumenow that A is a maximal order in its group of quotients AA−1. To provethat S is a maximal order, let h ∈ H = SS−1 and J an ideal of S such thathJ ⊆ J . Clearly, H = AA−1 oG. So h = a−1

1 a2g, for some a1, a2 ∈ A andg ∈ G and a−1

1 a2gJ ⊆ J . It is easily seen that J = (J ∩ A)G = G(J ∩ A),and thus a−1

1 a2J ⊆ J . Hence, a−11 a2(J∩A) ⊆ (J∩A). Since A is a maximal

order it follows that a−11 a2 ∈ A and thus h ∈ S, as desired. That Jh ⊆ J

implies h ∈ S is proved by a symmetric argument. Hence, S is a maximalorder.

More general, in [45, Lemma 10.5.2], the authors consider the semidi-rect product S = S1 o S2 of two monoids S1 and S2 determined by ahomomorphism σ : S2 → Aut(S1), and they prove that these constructionsyield classes of maximal orders. Recall that an ideal I of S1 is said to beσ-invariant if σ(s2)(I) = I for any s2 ∈ S2. A σ-invariant ideal Q of S1 issaid to be σ-prime if IJ ⊆ Q implies I ⊆ Q or J ⊆ Q, for any σ-invariantideals I, J of S1. Note that, in this case, QS is an ideal of S (for moredetails, see [45]).

Lemma 6.3.2. Let S1 and S2 be monoids satisfying the ascending chaincondition on one-sided ideals, each contained in a finitely generated tor-sion free abelian-by-finite group. Suppose σ : S2 → Aut(S1) is a monoidhomomorphism with σ(S2) finite. Let S = S1 oσ S2 be the correspondingsemidirect product. Then also S satisfies the ascending chain condition onone-sided ideals. If each Si is a maximal order, then S is a maximal orderin a torsion free abelian-by-finite group.

We give a concrete example illustrating Lemma 6.3.2. We will need thefollowing monoid of I-type, that is introduced and studied in [27], [45].



Example 6.3.3. The monoid M = 〈x, y | x2 = y2〉 is of I-type.

Proof. Let Fa2 = gr(a, b | ab = ba) denote the free abelian monoid of ranktwo with generators a and b. Let Sym2 = {1, σ} be the symmetric groupon {a, b}. So σ(a) = b and σ(b) = a. Let G = Fa2 oσ Sym2 denote thesemidirect product, where the action of Sym2 on Fa2 is defined naturallyvia the permutation σ. Thus

(c, α)(d, β) = (cα(d), αβ),

for c, d ∈ Fa2 and α, β ∈ Sym2.For c ∈ Fa2 we put σc = σ, if c has odd length and σc = 1, if c has even

length. Let FaM2 = 〈a, b〉 be the free abelian submonoid of Fa2 generatedby a and b. It easily is verified that

T = {(c, σc) | c ∈ FaM2}

is a submonoid of G. We claim that the monoid of I-type T is isomorphicwith S. Indeed, Let x1 = (a, σ) and x2 = (b, σ). A direct verification showsthat T = 〈x1, x2〉 and x2

1 = (ab, 1) = (ba, 1) = x22. Because of the left-right

dual it follows that x21 = x2

2 is the only defining relation for T . Hence, T isisomorphic with S, as desired.

Let D = 〈u1, u2, u3, u4 | u1u2 = u3u4〉 be the abelian Anderson monoid.Then, x and y of Example 6.3.3 act on D as follows:

ux1 = u2, u

x3 = u4, u

x2 = u1, u

x4 = u3, u

y1 = u3, u

y3 = u1, u

y2 = u4, u

y4 = u3.

Since the action of x and y preserves the relation u1u2 = u3u4, we canextend it to an action σ of M on D.

Example 6.3.4. ([45, Example 10.5.3]) The monoid

S = DoσM = 〈u1, u2, u3, u4, x, y | u1u2 = u3u4, x2 = y2, ux

1 = u2, ux3 = u4,

ux2 = u1, u

x4 = u3, u

y1 = u3, u

y3 = u1, u

y2 = u4, u

y4 = u3〉

is a maximal order satisfying the ascending chain condition on one-sidedideals and the minimal primes of S are P1 = (u1u2, u1u4, u2u3)S, P2 =(u1u2, u1u3, u2u4)S and P3 = (x2)S. Furthermore, for any field K, thesemigroup algebra K[S] is a maximal order.

Recall that another tool for constructing examples of semigroups thatare maximal orders is given by Proposition 4.2.1. Indeed, starting fromthe abelian maximal orders in Theorems 5.1.2 and 5.2.2, Proposition 4.2.1gives the key how one can construct new classes of non-abelian maximalorders, not only via semi-direct products.


Appendix A

Maximal order applicationsin Space-Time Coding

Quite recently one discovered that maximal orders are useful in codingtheory, especially in the space-time coding. Assume that information is sentfrom a source over a noisy channel to a receiver. A fundamental problemof coding theory is to determine what message was sent on the basis ofthe approximation that was received. The mathematical theory of error-correcting codes dates back to Shannon ([70]) who in 1948 published a veryremarkable paper on the existence of good error-correcting systems. Atthe same time Hamming ([34]) constructed perfect single-error-correctingcodes.

A space-time code (STC) is a method employed to improve the reliabil-ity of data transmission in wireless communication systems using multipletransmit antennas. Space-time codes rely on transmitting multiple, redun-dant copies of a data stream to the receiver in the hope that at least someof them may survive the physical path between transmission and receptionin a good enough state to allow reliable decoding.

In the next section we go more into detail about space-time codes andwe explain how maximal orders can be used to obtain new results in thecoding theory. For a survey about STC we refer the reader to [4, 5, 35, 66].We will mostly use the notation and terminology of [35].

127

A.1. Space-Time Coding 128

A.1 Space-Time Coding

As said before, multiple-antenna wireless communication promises veryhigh data rates, in particular when we have perfect channel state infor-mation available at the receiver. Since we will use n receive antennas and ntransmit antennas, a natural ambient space is the space Mn(C), of complexn× n matrices.

From the pairwise error probability point of view, i.e. since we wantto minimize the expecting value of the chance that we decode wrong, per-formance of a space-time code is dependent on two parameters: diversitygain and coding gain. Diversity gain is the minimum of the rank of thedifference matrix X −X

′taken over all distinct code matrices X,X

′ ∈ C,also called the rank of the code C. When C is full-rank, i.e. all matrices in Chave full rank or all matrices are invertible, the coding gain is proportionalto the determinant of the matrix (X −X

′)(X −X

′)H , where H indicates

the Hermitian transpose of a matrix. The minimum of this determinanttaken over all distinct code matrices is called the minimum determinantof the code C and is denoted by δmin(C). A good code will attempt tomaximize δmin(C).

The idea of using maximal orders then is that maximal orders providethe best codes in terms of minimum determinant versus average power. Theaverage power is the sum of the energies of all the code matrices dividedby the code size and the energy of a code word (= matrix) is the Frobeniusnorm of the matrix (i.e. the sum of the squares of all the matrix elements).So the study of maximal orders is clearly motivated by an analogy fromthe theory of error correcting codes: why would one use a particular codeof a given minimum distance and length, if a larger code with the sameparameters is available. We will explain more about the role of maximalorders, when we go to Theorem A.1.6.

For this we first briefly recall the basic definitions from cyclic algebrasand their use in ST-coding. In the following we will always consider numberfield extensions E/F , where F denotes the base field. We assume that E/Fis a cyclic field extension of degree n with cyclic Galois group Gal(E/F ) =gr(σ) (the number n will later be used as the number of transmit andreceive antennas). Let A = (E/F, σ, γ) be the corresponding cyclic algebraof degree n (n is also called the index of A), that is

A = E ⊕ uE ⊕ u2E ⊕ · · · ⊕ un−1E,

as a (right) vector space over E. Here u ∈ A is an auxiliary generatingelement subject to the relations xu = uσ(x) for all x ∈ E and un = γ ∈

Maximal order applications in Space-Time Coding

A.1. Space-Time Coding 129

F ∗. An element a = x0 + ux1 + · · · + un−1xn−1 ∈ A has the followingrepresentation as a matrix

A =

x0 γσ(xn−1) γσ2(xn−2) · · · γσn−1(x1)x1 σ(x0) γσ2(xn−1) γσn−1(x2)x2 σ(x1) σ2(x0) γσn−1(x3)...

...xn−1 σ(xn−2) σ2(xn−3) · · · σn−1(x0)

.

We refer to this as the standard matrix representation of A. An algebra Ais called simple if it has no nontrivial ideals. An F -algebra A is central ifits center Z(A) = F .

There is a nice link between cyclic algebras and central simple algebras(see [67]).

Proposition A.1.1. Any cyclic algebra is a central simple F -algebra.

A division algebra may be represented as a cyclic algebra in many waysas demonstrated by the following example concerning the Golden algebra.

Example A.1.2. The Golden algebra is related to the Golden numberθ = 1+

√5

2 , one of the roots of θ2−θ−1 = 0 (θ = 1−√

52 is the other one). It

turns out ([5]) that the division algebra A (to construct the Golden algebra)is a cyclic algebra with F = Q(i), E = Q(i,

√5), γ = i and σ(

√5) = −

√5.

So Gal(E/F ) = gr(σ) ∼= Z2, u2 = i, xu = uσ(x), for all x ∈ Q(i,√

5) and

A =(Q(i,

√5)/Q(i), σ, i

)∼= Q(i,

√5)⊕ uQ(i,

√5).

Furthermore elements of A can be written as 2× 2-matrices(x0 iσ(x1)x1 σ(x0)

),

for x0, x1 ∈ E = Q(i,√

5).

In the following example, we introduce natural orders.

Example A.1.3. In any cyclic algebra we can always choose the elementγ ∈ F ∗ to be an algebraic integer. We immediately see that the OF -module

Λ = OE ⊕ uOE ⊕ · · · ⊕ un−1OE ,

where OE is the ring of integers in E, is an OF -order in the cyclic algebra(E/F, σ, γ). We refer to this OF -order as the natural order. It will alsoserve as a starting point when searching for maximal orders.


A.2. Comments and further problems 130

To obtain a good code, one should try to maximize the density of thecode, or equivalent, one should try to minimize the fundamental parallelo-tope. Hence, deriving a lower bound for the measure of the fundamentalparallelotope is an interesting thing to do.

In [35], the authors derived such a lower bound in some particular casesand they showed that the discriminant is the tool for finding a lower boundfor the fundamental parallelotope. We only state here the results in caseF = Q(i).

Definition A.1.4. Let m = dimF A. The discriminant of the R-order Λis the ideal d(Λ/R) in R generated by the set

{det(tr(xixj))mi,j=1 | (x1, · · · , xm) ∈ Λm}.

Proposition A.1.5. Let F = Q(i), R = Z[i], and assume that Λ =(E/F, σ, γ) is an R-order. Then the measure of the fundamental paral-lelotope equals

m(Λ) = |d(Λ/R)|.

It turns out (see for example [67, Theorem 25.3]) that all the maximalorders in a division algebra share the same discriminant, so mostly one willrefer to this discriminant as the discriminant of the division algebra. Inthis sense a maximal order has the smallest possible discriminant amongall orders within a given division algebra, as all orders are contained in amaximal one.

Finally, in the following theorem, maximal orders are used to obtain alower bound for the measure of the fundamental parallelotope.

Theorem A.1.6. Let Λ be a maximal R-order (R = Z[i]) in a centraldivision algebra of index n over the field Q(i) (where n is the number oftransmit/receive antennas).

Then the measure of a fundamental parallelotope of the correspondinglattice

m(Λ) ≥ 10n(n−1)/2.

A.2 Comments and further problems

As maximal orders are interesting in space-time coding, one might askwhether there can be direct applications of our research in maximal ordersemigroup algebras.



In Chapter 6, we defined monoids of I-type S and we mentioned thattheir semigroup algebras K[S] are Noetherian domains, since their groupsof quotients are torsion-free (Theorems 2.3.3 and 2.3.4). Furthermore, weshowed in Theorem 6.1.2 that semigroup algebras K[S] of I-type satisfy theconditions of Theorem 4.1.3 and thus, in particular, they are PI Noetherianmaximal orders. By Theorem 2.5.6, it follows that such semigroup alge-bras are maximal R-orders (R = Z(K[S])) in the central division algebrasQcl(K[S]). But the central division algebra Qcl(K[S]) can not be of finiteindex over a number field, since a non-trivial semigroup of I-type is neverfinite. So a first question for further work might be what happens whenone drops the condition that the central division algebra is of finite indexover the field Q(i) in Theorem A.1.6. Even if this situation might not behelpful in practice, this question might be interesting from the theoreticalpoint of view.

On the other hand, in order to work with finite dimensional algebras,let us look at finite semigroups. Before we do this we give some moredefinitions. Recall that a semigroup with a zero θ is 0-simple if S2 = S,S2 6= {θ} and S has no ideals other than S and {θ}. A semigroup S iscompletely 0-simple if it is 0-simple and it contains a primitive idempotent,that is, it contains a minimal non-zero idempotent (an element e ∈ S whichsatisfies e = e2) with respect to the partial ordening e ≤ f . Recall thate ≤ f if and only if ef = fe = e. Let G be a group and I,M be non-empty sets. Let P = (pmi) be a M × I matrix with entries from G0. Forg ∈ G0, write (g)im, i ∈ I,m ∈ M for the I ×M matrix with (i,m)-entryg and all other entries θ. All matrices (θ)im are identified and it is calledthe zero matrix . These elements form a semigroup M0(G, I,M,P ) withmultiplication AB = A◦P ◦B where A,B ∈M0(G, I,M,P ) and ◦ denotesordinary multiplication of matrices. The matrix P is called the sandwichmatrix and we say that M0(G, I,M,P ) is a (Rees) matrix semigroup. Thematrix P is regular if it has at least one non-zero entry in every row andcolumn. When the sets I and M are finite, say | I |= n and |M |= m, wewill write M0(G, I,M,P ) simply as M0(G,n,m, P ).

The following theorem [15, Theorem 3.5, Theorem 3.9] characterizescompletely 0-simple semigroups.

Theorem A.2.1. A semigroup is completely 0-simple if and only if it isisomorphic to a matrix semigroup M0(G, I,M,P ) with regular sandwichmatrix P .

Let S be a completely 0-simple semigroup M0(G, I,M,P ). Then it iswell known that the contracted semigroup algebra K0[S] is isomorphic to



the ring of matrix type M(K[G], I,M, P ), where we regard the entries inP as elements of the group algebra K[G] (see [60, Lemma 5.1]).

It is known by the Theorem of Maschke [17, 3.14] when a group algebraof a finite group is semisimple. This property for semigroup algebras ofcompletely 0-simple semigroups is characterized in the following theorem.

Theorem A.2.2. Let S = M0(G,n,m, P ) be a finite completely 0-simplesemigroup and K a field. Then K0[S] is semisimple if and only if

1. the characteristic of K does not divide |G|;

2. P is invertible (in particular m = n) regarded as a matrix over K[G].

Hence, in order to find more maximal orders to obtain applications inspace-time coding, a first step could be to look at maximal orders withincontracted semigroup rings of finite completely 0-simple semigroups.


Samenvatting

Het doel van het werk in deze thesis is het presenteren van nieuwe klassenvan algebra’s met een rijke algebraısche structuur, zoals priem Noethersemaximale orders. Door de rol van Noetherse ringen in de algebraısche be-nadering van niet-commutatieve meetkunde, verdienden concrete klassenvan eindig voortgebrachte algebra’ s recent de aandacht. Dankzij toepas-singen in oplossingen van de Yang-Baxter vergelijking, wakkerden dezeklassen ook de interesse aan in andere onderzoeksdomeinen, zoals de wiskun-dige fysica. Mede daardoor werden deze klassen reeds uitgebreid bestudeerd,zie bijvoorbeeld [20, 21, 25, 27, 39, 68].

Wij zullen in deze thesis constructies geven van eindig gepresenteerdealgebra’s over een lichaamK, die gedefinıeerd zijn door monomische relaties(zie onder andere de motiverende voorbeelden 3.1.1, 3.1.2 and 3.1.3). Omte kunnen bepalen wanneer deze algebra’s Noetherse maximale orders zijn,tonen wij aan dat de onderliggende homogene informatie de algebraıschestructuur van de algebra bepaalt. Bijgevolg, is het natuurlijk deze algebra’ste beschouwen als semigroepsalgebra’s en de structuur van deze monoıde Ste onderzoeken. Bovendien volgt er dat vele van deze algebra’s deelalgebra’szijn van groepsalgeba’s van polycyclische-bij-eindige groepen. Deze linknaar groepsalgebra’s, die zeer interessant is, vormt een rijke bron van con-structies, niet alleen voor de niet-commutatieve ringtheoristen, maar ookvoor onderzoekers in semigroeptheorie en bepaalde aspecten uit groepen-theorie.

In het algemeen blijft het een onopgelost probleem te karakteriserenwanneer een willekeurige semigroepsalgebra Noethers is en wanneer heteen priem Noethers maximaal order is. Zelfs voor groepsalgebra’s is ditprobleem onopgelost. De enige klasse van groepen waarvoor een positiefantwoord gegeven is, is de klasse van de polycyclische-bij-eindige groepen,er is namelijk een conjectuur gesteld dat deze groepen de enige groepenzijn die een Noetherse groepsalgebra geven.

133

134

Brown ([6, 7]) karakteriseerde wanneer zulke groepsalgebra’s Noethersemaximale orders zijn, en wanneer unieke factorisatie ringen, zoals bestudeerddoor Chatters en Jordan ([12]).

Bijgevolg, wanneer we bovenstaand probleem willen onderzoeken voorsemigroepsalgebra’s, is het in eerste instantie natuurlijk deelmonoıdes vanpolycyclische-bij-eindige groepen te beschouwen. In [45] karakteriseerdenJespers en Okninski wanneer semigroepsalgebra’s van zulke semigroepenrechts Noethers zijn (Stelling 3.1.4). Uit de karakterisatie volgt dat dezesemigroepsalgebra’s rechts Noethers zijn enkel en alleen indien zij linksNoethers zijn en dat, in dit geval, de semigroep S eindig voortgebracht is.In het algemeen is dit niet altijd het geval (zie [60, Theorem 6]). Alhoewel,indien S een abelse monoıde is, bewees Gilmer [28] dat de semigroepsalgebraK[S] Noethers is enkel en alleen indien S eindig voortgebracht is.

Wat betreft het maximale order probleem, als S een abelse monoıdeis, bewees Anderson [1, 2] dat K[S] een priem Noethers maximaal or-der is enkel en alleen indien S een deelmonoıde is van een eindig voort-gebrachte torsie-vrije abelse groep en S een maximaal order is in zijn groepvan quotienten. Meer algemeen, bewees Chouinard [14] dat een commu-tatieve monoıde algebra K[S] een Krull domein is enkel en alleen indienS een deelmonoıde is van een torsie-vrije abelse groep die voldoet aan destijgende ketenvoorwaarde op cyclische deelgroepen en S is een Krull orderin zijn groep van quotienten. In het bijzonder volgt er dat de hoogte eenpriemen van K[S], die voortkomen van minimale priemen in S, cruciaalzijn en dat de klassegroep van K[S] isomorf is met de klassegroep van S.

Buiten de abelse gevallen waarover we het hierboven hadden, was ertot voor kort alleen maar een antwoord op het probleem bekomen voorspeciale gevallen van cancellatieve semigroepen, zoals deelmonoıdes vantorsie-vrije eindig voortgebrachte abels-bij-eindige groepen ([41]), of voorspeciale klassen van maximale orders, zoals hoofdideaalringen ([38, 49]) enunieke factorisatie ringen ([47, 48]).

In deze thesis zetten wij dit onderzoek verder, wij onderzoeken namelijkwanneer een semigroepsalgebra van een deelmonoıde van een polycyclische-bij-eindige groep een priem Noethers maximaal order is en wij beschrijvenhoogte een priemidealen van zulke semigroepsalgebra’s. Dus geven wij, vooreen grote klasse van cancellative semigroepen een antwoord op bovenstaandopen probleem.

We beschrijven nu kort de inhoud per hoofdstuk. Hoofdstuk 2 is toege-voegd voor het gemak van de lezer. Dit hoofdstuk bevat de belangrijk-ste resultaten en definities van semigroepen ringtheorie die we nodig

Nederlandstalige Samenvatting

135

hebben in de thesis. In Hoofdstuk 3 brengen wij verslag uit over de resul-taten die wij bekomen zijn door samenwerking met Jespers en Okninski enwij karakteriseren, onder een invariantie voorwaarde op minimale priemen,Noetherse semigroepsalgebra’s die priem maximale orders zijn, voor deel-monoıdes van polycyclische-bij-eindige groepen (Stelling 3.4.3). Kennisover de priemidealen blijkt weer cruciaal te zijn en daarom beschrijven wijeerst de hoogte een priemidealen van zulke semigroepsalgebra’s met K[S]priem (Stelling 3.2.2). Als K[S] een domein is, d.w.z. als SS−1 torsie-vrijis, dan is dezelfde informatie over hoogte een priemen, en zelfs over priemendie niet van hoogte een zijn, bekomen door Jespers en Okninski in [45]. Indeze thesis tonen wij echter dat de informatie over priemen, die niet vanhoogte een zijn, niet meer geldig is indien K[S] priem is en geen domein(Voorbeeld 4.3.2).

Als een gevolg van de beschrijving van de hoogte een priemen, tonenwij going up en going down eigenschappen aan tussen priemidealen van Sen priemidealen van S∩H, waarbij H een deelgroep is van eindige index inG = SS−1 (Stelling 3.2.5). Deze resultaten zijn analoog aan de belangrijkeresultaten die geweten zijn over het gedrag van priemidealen tussen eenring die gegradeerd is door een eindige groep en zijn homogene componentvan graad e (de identiteit van de graderende groep). Als toepassing tonenwij dat de klassieke Krull dimensie van K[S], clKdim(K[S]), de som is vande priem dimensie van S en de plinth lengte van de eenhedengroep U(S)(Stelling 3.3.1). Tevens breiden wij een resultaat van Schelter uit naar hetmonoıde niveau: de priemdimensie van S is de som van de hoogte en dediepte van elk priemideaal van S (Stelling 3.3.3).

In Hoofdstuk 4 beschrijven wij, als een gevolg van de bekomen karakteri-satie in Hoofdstuk 3, wanneer een semigroepsalgebra van een deelmonoıdevan een polycyclische-bij-eindige groep een priem Noethers maximaal orderis die voldoet aan een polynoomidentiteit, dit betekent dat wij karakteri-seren wanneer de semigroepsalgebra van een deelmonoıde van een eindigvoortgebrachte abels-bij-eindige groep een priem Noethers maximaal orderis (Stelling 4.1.3). Deze karakterisatie is equivalent met het eerdere resul-taat Stelling 3.1.6, indien de quotientengroep torsie-vrij is. Bovendien tonenwij aan dat, als de polycyclische-bij-eindige groep in Stelling 3.4.3 abels-bij-eindig is, dat dan de invariantie voorwaarde op de minimale priemenin Stelling 3.4.3 niet alleen voldoende is, maar ook nodig. In Sectie 4.2illustreren wij Stelling 4.1.3 met een voorbeeld van een priem Noethersemaximale order semigroepsalgebra (Voorbeeld 4.3.2).


136

Dit voorbeeld is tevens ook het voorbeeld dat aantoont dat de informatieover priemidealen niet meer geldt indien de hoogte van het priemideaalstrict groter is dan een (zie hierboven). De bekomen resultaten in de hoofd-stukken 3 en 4 kunnen gevonden worden in de artikels [30] en [31].

Stelling 4.1.3 herleidt het probleem van het beschrijven wanneer K[S]een priem Noethers maximaal order is naar de algebraısche structuur van S.Bijgevolg levert deze stelling een belangrijk hulpmiddel om nieuwe klassenvan zulke algebra’s te construeren. Voor sommige voorbeelden kunnen denodige voorwaarden op S gemakkelijk geverifieerd worden, maar voor an-dere voorbeelden echter, blijft er nog steeds wat substantieel werk te doen.Om deze reden ontwikkelden wij een nuttig algemeen resultaat om nietabelse deelmonoıdes van abels-bij-eindige groepen die maximale orders zijnte construeren, beginnende van abelse maximale orders (Propositie 4.2.1).

Anderson, Gilmer en Chouinard karakteriseerden abelse eindig voortge-brachte monıdes die maximale orders zijn ([1, 2, 14, 28]). Maar buiten dezeaangename en nuttige structurele karakterisatie, blijft het volgende pro-bleem een uitdagend probleem voor een eindig voortgebrachte deelmonoıdeS van een torsie-vrije abelse groep: bepaal nodige en voldoende voorwaar-den op de definierende relaties van S opdat S een maximaal order is. InHoofdstuk 5, lossen wij dit probleem op voor monoıdes die gedefinieerdzijn door een en twee relaties (Stelling 5.1.2, 5.2.2). Dus, in tegenstellingtot de structurele beschrijving die bekomen is door Anderson, Gilmer enChouinard, steunt onze bijdrage om een beschrijving van maximale ordersvia monomische relaties te bekomen, meer op een computationele benade-ring (gebaseerd op presentaties). Het voordeel is dat deze denkwijze ons ooktoelaat de klassegroepen cl(S) van zulke monoıdes S, en dus ook van hunalgebra’s K[S], te berekenen (Stelling 5.1.4, 5.2.4). Deze groep is het ba-sis hulpmiddel in de studie van arithmetische eigenschappen van maximaleorders ([23]) en bijgevolg volgt er dat onze bekomen resultaten over abelsemaximale orders en klassegroepen, die opgeschreven zijn in het artikel [32],ook interessant kunnen zijn voor experten in verwante onderzoeksdomeinen.

In Hoofdstuk 6 illustreren wij de theorie die ontwikkeld is in de eerderehoofdstukken op verschillende concrete klassen van voorbeelden van semi-groepsalgebra’s en we komen terug op de niet-commutatieve algebra’s dieverscheidene eigenschappen delen met polynoom algebra’s in eindig veelcommuterende variabelen, waarover wij het in het begin van deze samen-vatting hadden.


137

Wij beschouwen namelijk monoıdes van I-type, die geıntroduceerd zijndoor Gateva-Ivanova en Van den Bergh in [27], en later ook bestudeerdzijn door Jespers en Okninski in [43]. Wij herbewijzen, met behulp vanStelling 4.1.3, dat deze semigroepsalgebra’s altijd Noetherse maximale or-ders en domeinen zijn. Bovendien veralgemenen wij het concept van eenmonoıde van I-type naar een monoıde van IG-type en wij karakteriseren,opnieuw met behulp van Stelling 4.1.3, wanneer deze semigroepsalgebra’spriem Noetherse maximale orders zijn. Al de resultaten omtrent monoıdesvan IG-type zijn gepubliceerd in ons artikel [29]. Tevens beschrijven wij inHoofdstuk 6 een paar andere voorbeelden van maximale orders.

In de Appendix, ten slotte, geven wij commentaar op toepassingen vanmaximale orders in de space-time coding. We beschrijven kort de tamelijkrecent ontwikkelde theorie van space-time codes en we verklaren hoe maxi-male orders gebruikt kunnen worden om nieuwe resultaten te ontwikkelen indit onderzoeksdomein. Deze toepassingen wijzen erop dat maximale ordersmisschien niet enkel interessant kunnen zijn voor experten in de algebra,maar misschien ook voor specialisten in de codetheorie.


138


Bibliography[1] Anderson D.F., Graded Krull domains, Comm. Algebra 7 (1979), 79–

106.

[2] Anderson D.F., The divisor class group of a semigroup ring, Comm.Algebra 8(5) (1980), 467–476.

[3] Alamouti S.M., A simple transmit diversity technique for wireless com-munications, IEEE Journal on Selected Areas in Communications 16(8) (1998), 1451–1458.

[4] Belfiore J.C., Oggier F., Rekaya G. and E. Viterbo, Perfect Space-Time Block Codes, IEEE Trans. Inf. Theory, vol. 52 (2006), 3885–3902.

[5] Belfiore J.C., Rekaya G. and Viterbo E., The Golden Code: a 2 × 2full rate space-time code with non-vanishing determinant, IEEE Trans-actions on Information Theory, Vol 51, n.4 (2005), 1432–1436.

[6] Brown K.A., Height one primes of polycyclic group rings, J. LondonMath. Soc. 32 (1985), 426–438.

[7] Brown K.A., Corrigendum and addendum to ‘Height one primes ofpolycyclic group rings’, J. London Math. Soc. 38 (1988), 421–422.

[8] Cedo F., Jespers E. and Okninski J., Semiprime quadratic algebras ofGelfand-Kirillov dimension one. J. Algebra Appl. 3 (2004), 283–300.

[9] Chamarie M., Anneaux de Krull non commutatifs, These, UniversiteClaude-Bernard - Lyon I, 1981.

[10] Chamarie M., Anneaux de Krull non commutatifs, J. Algebra 72(1981), 210–222.

[11] Chapman S.T., Garcia-Sanchez P.A., Llena D. and Rosales J.C., Pre-sentations of finitely generated cancellative abelian monoids and non-negative solutions of systems of linear equations, [J] Discrete Appl.Math. 154 (2006), 1947–1959.

139

140

[12] Chatters A.W. and Jordan D.A., Non-commutative unique factorisa-tion rings, J. London Math. Soc. (2) 33 (1986), 22–32.

[13] Chin W. and Quinn D., Rings graded by polycyclic-by-finite groups,Proc. Amer. Math. Soc. 102 (1988), 235–241.

[14] Chouinard II L.G., Krull semigroups and divisor class groups, Canad.J. Math. 23 (1981), 1459–1468.

[15] Clifford A.H. and Preston G.B., The Algebraic Theory of Semigroups,Vol.I, Amer.Math.SOC., Providence, 1961.

[16] Cohen M. and Rowen L., Group graded rings, Comm. Algebra 11(1983), 1253–1270.

[17] Curtis C.W and Reiner I., Representation Theory Of Finite GroupsAnd Associative Algebras, John Wiley and Sons, New York, 1962.

[18] De Kimpe K., Almost Bieberbach groups: Affine and PolynomialStructures, Lecture Notes in Mathematics 1639, Springer (1996).

[19] Divinsky N.J., Rings And Radicals, University of Toronto Press,Toronto, 1965.

[20] Etingof P., Guralnick R. and Soloviev A., Indecomposable set-theoretical solutions to the quantum Yang-Baxter equation on a setwith a prime number of elements, J. Algebra 249 (2001), 709–719.

[21] Etingof P., Schedler T. and Soloviev A., Set-theoretical solutions of thequantum Yang-Baxter equation, Duke Math. J. 100 (1999), 169–209.

[22] Faith C., Algebra II, Ring Theory, Springer-Verlag, New York, 1976.

[23] Fossum R., The Divisor Class Group of a Krull Domain, Springer-Verlag, New York, 1973.

[24] Garcia-Sanchez P.A. and Rosales J.C., Finitely Generated AbelianMonoids, Nova Science Publishers, New York, 1999.

[25] Gateva-Ivanova T., A combinatorial approach to the set-theoretic so-lutions of the Yang-Baxter equation, J. Math. Phys. 45 (2004), 3828–3858.

[26] Gateva-Ivanova T., Jespers E. and Okninski J., Quadratic algebras ofskew type and the underlying monoids, J. Algebra 270 (2003), 635-659.

[27] Gateva-Ivanova T. and Van den Bergh M., Semigroups of I-type, J.Algebra, 206 (1998), 97-112.

Bibliography

141

[28] Gilmer R., Commutative semigroup rings, Univ.Chicago Press,Chicago, 1984.

[29] Goffa I. and Jespers E., Monoids of IG-type and maximal orders, J.Algebra 308 (2007), 44–62.

[30] Goffa I., Jespers E. and Okninski J., Primes of height one and a classof Noetherian finitely presented algebras, Internat. J. Algebra Comput.Vol.17, No.7 (2007), 1465–1491.

[31] Goffa I., Jespers E. and Okninski J., Semigroup algebras of submonoidsof polycyclic-by-finite groups and maximal orders, to appear in Alge-bras Repres. Theory.

[32] Goffa I., Jespers E. and Okninski J., Integrally closed domains withmonomial presentations, submitted.

[33] Goodearl K.R. and Warfield R.B., An Introduction to NoncommutativeNoetherian Rings, Cambridge Univ. Press, New York, 1989.

[34] Hamming R.W., Error detecting and error correcting codes, Bell Sys-tem Tech. J. 29, (1950) 147–160.

[35] Hollanti C., Lahtonen J., Ranto K. and Vehkalahti R., On the dens-est MIMO lattices from cyclic division algebras, submitted to IEEETransactions on Information Theory.

[36] Jespers E., Krempa J. and Puczylowski E.R., On radicals of gradedrings, Comm. Algebra 10 (1982), 1849–1854.

[37] Jespers E. and Okninski J., Nilpotent semigroups and semigroup alge-bras, J. Algebra 169 (1994), no. 3, 984–1011.

[38] Jespers E. and Okninski J., Semigroup algebras that are principal idealrings, J. Algebra 183 (1996), 837-863.

[39] Jespers E. and Okninski J., Binomial semigroups, J. Algebra 202(1998), 250–275.

[40] Jespers E. and Okninski J., Semigroup algebras and maximal orders,Canad. Math. Bull. Vol. 42 (3) (1999), 299–306.

[41] Jespers E. and Okninski J., Semigroup algebras and Noetherian maxi-mal orders, J. Algebra 238 (2001), 590–622.

[42] Jespers E. and Okninski J., Submonoids of polycyclic-by-finite groupsand their algebras, Algebras Repres. Theory 4 (2001), 133–153.

[43] Jespers E. and Okninski J., Monoids and groups of I-type, AlgebrasRepres. Theory 8 (2005), 709-729.

Bibliography

142

[44] Jespers E. and Okninski J., Noetherian semigroup algebras of sub-monoids of polycyclic-by-finite groups, Bull. London Math. Soc. 38(2006), 421–428.

[45] Jespers E. and Okninski J., Noetherian Semigroup Algebras, Algebraand Applications vol.7, Springer, 2007.

[46] Jespers E. and Wang Q., Hereditary semigroup algebras, J. Algebra233 (2000), no. 1, 409–424. (Correction by the editor) and J. Algebra229 (2000), 532–546.

[47] Jespers E. and Wang Q., Noetherian unique factorization semigroupalgebras, Comm. Algebra 29 (2001), 5701–5715.

[48] Jespers E. and Wang Q., Height one prime ideals in semigroup algebrassatisfying a polynomial identity, J. Algebra 248 (2002),118–131.

[49] Jespers E. and Wauters P., Principal ideal semigroup rings, Comm.Algebra 23 (1995), 5057-5076.

[50] Kargapolov M.I. and Merzljakov Ju.I., Fundamentals of the Theory ofGroups, Springer-Verlag, New York, 1979.

[51] Krause G.R. and Lenagan T.H., Growth of Algebras and Gelfand-Kirillov Dimension, Revised edition. Graduate Studies in Mathemat-ics, 22. American Mathematical Society, Providence, RI, 2000.

[52] Lebruyn L., Van den Bergh M. and Van Oystaeyen F., Graded Orders,Switzerland Birkhauser Verlag AG, 1988.

[53] Letzter E.S. and Lorenz M., Polycyclic-by-finite group rings are cate-nary, Math. Res. Letters 6 (1999), 183194.

[54] Malcev A.I., Nilpotent Semigroups, Uc. Zap. Ivanovsk. Ped. Inst. 4(1953), 107–111. (In Russian)

[55] Marubayashi H., Noncommutative Krull rings, Osaka J. Math. 12(1975), 703–714.

[56] Marubayashi H., On bounded Krull prime rings, Osaka J. Math. 13(1976), 491–501.

[57] Marubayashi H., A characterization of bounded Krull prime rings,Osaka J. Math. 15 (1978), 13–20.

[58] McConnell J.C. and Robson J.C., Noncommutative Noetherian Rings,Wiley, New York, 1987.

[59] B.H. Neumann and Tekla Taylor, Subsemigroups of nilpotent groups,Proc. Roy. Soc, Ser.A 274 (1963), 1–4.

Bibliography

143

[60] Okninski J., Semigroup Algebras, Marcel Dekker, New York, 1991.

[61] Okninski J., In search for Noetherian algebras, in: Algebra – Repre-sentation Theory, NATO ASI, pp. 235–247, Kluwer, 2001.

[62] Okninski J., Prime ideals of cancellative semigoups, Comm. Algebra32 (2004), 2733–2742.

[63] Okninski J., Nilpotent Semigroup Of Matrices, Math. Proc. Camb.Phil. Soc. 120 (1996), no. 4, 617–630.

[64] Passman D.S., The Algebraic Structure of Group Rings, Wiley, NewYork, 1977.

[65] Passman D.S., Infinite Crossed Products, Academic Press, New York,1989.

[66] Rajan B.S., Sethuraman B.A. and Shashidhar V., Full-Diversity, High-Rate Space-Time Block Codes From Division Algebras, IEEE Trans.Inf. Theory, vol. 49 (2003), 2596–2616.

[67] Reiner I., Maximal orders, Academic Press, New York 1975.

[68] Rump W., A decomposition theorem for square-free unitary solutionsof the quantum Yang-Baxter equation, Adv. Math. 193 (2005), 40–55.

[69] Segal D., Polycyclic Groups, Cambridge Univ. Press, Cambridge, 1983.

[70] Shannon C. E., A mathematical theory of communication, Bell SystemTech. J. 27 (1948), 379–423, 623–656.

[71] Wauters P., On some subsemigroups of noncommutative Krull rings,Comm. Algebra 12 (1984), 1751–1765.

Bibliography

144

Bibliography

Index

CR(I), 19D(S), 26F+, 65G+, 16I(ρ), 9J(R), 12P (S), 26S/ρ, 9S0, 7S1, 7X1(R), 12X1(S), 11cl(S), 26∆(G), 16∆+(G), 16GK(R), 13Spec(R), 12U(S), 6clKdim(R), 13dim(S), 11gr(T ), 8ht(P ), 12〈T 〉, 8B(R), 12M0(G, I,M,P ), 126MX(H), 32Z(S), 7pl(G), 18ρJ , 10Hsupp, 67supp, 7laR(X), 19

algebraaffine, 14central, 124cyclic, 123polynomial identity or PI, 13simple, 12, 124

Anderson monoid, 54annihilator, 18Asano order, 22

centre, 7congruence relation, 9

discriminant, 125domain, 12

finite conjugacy centre, 16finite conjugacy group, 17

Goldie’ s theorem, 20group, 6

abelian-by-finite, 10dihedral free, 24FC-centre, 16group of I-type, 103group of IG-type, 105P-by-Q, 10poly-infinite-cyclic, 16polycyclic-by-finite, 16residually finite, 43torsion-free, 16

Hirsch rank, 16

145

146

idealσ-invariant, 120σ-prime, 120completely prime, 19divisorial, 22essential, 20fractional, 22

integral, 22invertible, 22nil, 12nilpotent, 12

integral domain, 21integrally closed, 22

completely, 21

Krull order, 22divisor group, 23normalizing class group, 23

Levitzki-theorem, 12

matrix semigroupsandwich matrix, 126

regular, 126zero matrix, 126

maximal order, 21module

semisimple, 12simple, 12

monoid, 6Anderson, 54of I-type, 102of IG-type, 104

monomialrelations, 28semigroup, 32

Noetherian, 11normal element, 12

order

maximal, 21natural, 125

Ore set, 19

plinth, 17centric, 17plinth length, 18plinth series, 18

polynomial identity or PI, 13prime ideal, 12

going up and going down, 15lying over, 15

principal ideal theorem, 12

R-order, 24maximal, 24

radicalJacobson, 12prime or Baer, 12

regular element, 19ring

Artinian, 11catenary, 14classical Krull dimension, 13classical ring of quotients, 19Gelfand-Kirillov-dimension, 13Goldie dimension, 20Goldie ring, 20graded, 14localization, 19maximal order, 21Noetherian, 11normal element, 12order, 21prime, 12prime ideal, 12

height, 12minimal prime, 12

prime spectrum, 12semilocal, 12

Index

147

semiprime, 12semiprime ideal, 12semisimple, 12simple, 12strongly graded, 14uniform dimension, 20unique factorization domain,

23unique factorization ring, 22

Schelter’s theorem, 14semigroup

0-simple, 8completely, 126

cancellative, 10congruence relation, 9dimension, 11homomorphism, 8ideal, 8

divisorial, 25fractional, 25invertible, 25nil, 8nilpotent, 8

identity, 6integrally closed, 25

completely, 25Krull order, 26

normalizing class group, 26maximal order, 25monoid of I-type, 102monoid of IG-type, 104monomial, 32nil, 8nilpotent, 8non-degenerate, 102null, 6Ore condition, 10prime, 11prime ideal, 11

height, 11minimal prime, 11

prime spectrum, 11rank, 11Rees factor, 9simple, 8subgroup, 7submonoid, 7subsemigroup, 7unit group, 6zero, 6

semigroup algebra, 7contracted, 8

semigroup ringcontracted, 8

space-time coding, 122coding gain, 123diversity gain, 123minimum determinant, 123

support, 7

theoremGoldie, 20Levitzki, 12principal ideal theorem, 12Schelter, 14

Yang-Baxter equation, 102

Index

noetherian semigroup algebras and prime maximal orders

Documents