some aspects of ring theory

SOME ASPECTS OF RING THEORY

A. W. GOLDIE

Ring theory arises from two main sources;

(i) associative algebras, especially group-algebras, and matrix theory,

(ii) polynomial rings, algebraic number theory and algebraic geometry.

The first of these sources led to the development of artinian rings which ischaracterised by its non-commutative nature and develops by the methods of thetheory of ideals and modules. The second led to commutative noetherian rings whichhas been studied by more basic and natural methods permitted by the commutativelaw of multiplication, together with these of ideal and field theory. At first the twodevelopments were wholly separate but have drawn together in the last decade,because of recognition of the universal value of the theory of modules and thegathering strength of the non-commutative theory. Since an artinian ring isnoetherian, it is natural enough to examine possible properties of noetherian ringswhich generalise in an illuminating way suitable parts of both the artinian theoryand the commutative theory. The search for results of this type has already broughtto light new classes of rings and revitalised some of the older results, especially inartinian rings.

1. Domains, prime and semi-prime rings

The technique of forming a ring of quotients is extensively employed in com-mutative ring theory, but is not used in artinian rings because nothing is gained, theyare already quotient rings. A non zero-divisor is already a unit of the ring.

In Van der Waerden [116], the first edition of 1930, the author asked whether anon-commutative integral domain can be embedded in a division ring, but theedition of 1937 carries a reference to the famous counter-example of Malcev [85].In the meantime Wedderburn [117] had shown that the quotient method workedfor euclidean domains and in Ore [92] there appeared the condition which is givenbelow. Some special cases of polynomial domains were discussed in Littlewood[81].Interest waned until Tamari [112] showed in 1953 that a Birkhoff-Witt algebra(this is a noetherian domain) has a quotient division ring. The general problem wassettled in Goldie [37]. These papers study the quotient problem only, the generalquestion of embeddability being a different matter; the reader is referred to B. H.Neumann [91] and P. M. Cohn [18].

Let R be a ring. An element of R is regular if it is neither a left nor right zerodivisor. A set # c R of regular elements is a right Ore set if any pair aeR,

Received 14 February, 1969.

[BULL. LONDON MATH. SOC, 1 (1969), 129-154]

BULL. 2

130 A. W. GOLDIE

gives rise to another pair ateR, q e # such that

ac{ = cav

A ring Q is a right quotient ring of R with respect to # provided that

0) 2 ^ ;(ii) the elements of # are units in Q;

(iii) the elements of Q have the form ac~1-, aeR, c e # .

Q exists if and only if ^ satisfies the right Ore condition, this result is due to Orefor domains and to Asano generally; see Jacobson [59]. The necessity of the Orecondition is clear, since c~la must be expressible in the form at c^1. Its sufficiencyis usually established by an elaborate form of the commutative procedure usingclasses of ordered pairs (a, c), but there is a different approach in Lambek [74Chapter 2], with the help of partial endomorphisms.

The existence of a quotient ring depends on the rank of the ring. A ring R hasfinite (right) rank when it has no infinite direct sums of non-zero right ideals. Aright ideal U of R is uniform if U ^ 0 and Ui n U2 i1 0 for any pair of non-zeroright ideals Uu U2 which lie in U.

Each non-zero right ideal / contains a uniform right ideal. Suppose not, then Rhas a sequence of non-zero right ideals {/„,/„'} with the properties:

Now 11 ®I2' © ... is an infinite direct sum, the existence of which contradicts theassumption that R has finite rank.

Let 17! © ... © Um and V1 © ... © Vn be direct sums of uniform right ideals ofR, each having maximal length. Each sum is an essential right ideal, it has non-zerointersection with every non-zero right ideal of R. In more usual terms, R is anessential extension of each sum, considered as a right /^-module. Now m = n, whichwas proved in Goldie [37] by a procedure following that in the Steinitz exchangetheorem. The invariant integer m is the (right) rank of R. The term infinite rightrank covers the case where R has infinite direct sums of right ideals.

THEOREM (1.1). The (right) rank of an integral domain R takes only the values 1 andoo. R has a right quotient ring if and only if the rank is 1.

For the satisfaction of the Ore condition is equivalent to R itself being a uniformright ideal, in other words, R has rank 1. When R has finite rank it contains auniform right ideal, and because R is a domain, one readily verifies that R itself isa uniform right ideal.

The requirement of finite rank is a weak sort of maximum condition on ideals,it is vacuous for commutative domains. An artinian domain is already a divisionring.f Certainly we can say that a right noetherian domain has rank one. This

t By an artinian ring we mean either a right or left (or both) artinian ring. The precise gloss isunnecessary in the general text.

SOME ASPECTS OF RING THEORY 131

leads to the important case of Birkhoff-Witt algebras or universal enveloping algebras.Let L be a finite-dimensional Lie algebra over a field K and U(L) be its universal

enveloping algebra, see Jacobson [61; Chapter 5]. Choose a basis uu ...,un of Lover K and suppose that in L

fc = l

U(L) is isomorphic to the ring of polynomials over K in indeterminates xl5 ...,xn

subject to the equationsn

Y \" —̂ "V Y — \ ' cti y * 7 7 . ^ 1 %>t•*,- -A-j •*•_/ - * i /_, / ijk •A'k, I, J 1 , . . . , n .

U(L) is a right and left noetherian domain and thus has right and left quotient rings;these are isomorphic division rings.

U(L) exhibits the better sort of behaviour, but the following instance is different.

Example. Let F be a field with an isomorphism a -> a which is not an auto-morphism and F be the subfield of images. Let F[x, — ] be the ring of polynomialsof the form

a0 + xax +... + xk ak, k ^ 0, at e F.

Multiplication follows the ruleax = xa.

The right ideals of this domain are principal; it has a right quotient ring, but nota left quotient ring. The left Ore condition breaks down because F ^ F. This isworth noting as a principal right ideal ring which is not left noetherian.

Simple noetherian rings have never been determined and results obtained forsimple rings usually extend to a wider context. The following is an example of asimple domain of the easiest one-sided ideal form but not a division ring.

Example. Let F(£) be the ring of rational functions in £. over a field F of zerocharacteristic and take F [x, £] to be the ring of differential polynomials in t overF(^). Its elements are subject to the rule

at = t(x+du/d£; aeF(^).

This is an integral domain, its right (and left) ideals are all principal. It is a simplering (no proper two-sided ideals).

The ideas developed so far for domains are now applied to a wider class of rings.A ring R is semi-prime if it has no nilpotent ideals. A ring R is a prime ring if thecondition aRb = 0, where a,beR, implies that either a = 0 or b = 0. An ideal Aof a ring I? is a semi-prime (prime) ideal when the factor ring R/A is a semi-prime(prime) ring.

A prime ring is a semi-prime ring. A commutative prime ring is an integraldomain and the semi-prime case has little interest. An artinian semi-prime ring issemi-simple and if prime is a simple ring, which is a full matrix ring over a divisionring.

132 A. W. GOLDIE

The following theorem is due to Goldie [38]. It had been proved in [37] forprime rings under two-sided conditions and in Lesieur-Croisot [76] for prime ringsunder one-sided conditions. Later proofs appear in Gabriel [33], Lambek [74],Procesi-Small [99].

THEOREM (1.2). A ring R has a right quotient ring Q which is a semi-simple artinianring if and only if

(1) R is a semi-prime ring,

(2) R has finite right rank,

(3) JR has maximum condition on right annihilators.

We take the present opportunity of inflicting another proof on the reader; ithas at any rate the merit of brevity. The third condition needs explanation. Let Sbe a subset of R. Denote

r(S) = (xeR; Sx = 0).

Then r(S) is a right ideal, the right annihilator of S. Left annihilators are similarlydefined, a typical one is

An interesting ideal in any ring R is the set

Z(R) = (zeR; r(z) is an essential right ideal).

Z(R) is a two-sided ideal of R. In the prescence of the maximum condition on rightannihilators Z(R) becomes a nil ideal. The concept is due to R. E. Johnson [63].

LEMMA (1.3). Let Rbe a ring satisfying conditions (1) and (3). Then a nil right(left) ideal of R is zero.

Proof. As aR is nil if and only if Ra is nil, we consider Ra ^ 0. The set ofright annihilators r(za), z ranging over R with za # 0, has maximal elements. Letr(b) be maximal, b = ta. Suppose that (yb)k = 0, (yb)11'1 ^ 0, where yeR, andk > 1. Then r(b) = r((j;Z>)'c~1)) by the maximality and hence byb = 0. ThusbRb = 0 and (Rb)2 = 0, b = 0, since R has no nilpotent left ideals. This contra-diction shows that Ra = 0 and the lemma follows.

LEMMA (1.4). Let R be any ring which satisfies conditions (2) and (3). For eachaeR there exists n > 0 such that a"R+r(a") is an essential right ideal.

Proof. Because of (3) there is an n ^ 1 such that r(o") = r(an+1). Thena"R n r(a") = 0. Let / be a right ideal and suppose that

I n (cfR+ria")) =0 .

The sum I+anI + a2nI+... is direct and, because R has finite rank, we concludethat 7 = 0.

Lemma (1.3) is due to Utumi [114] as regards proof and Lemma (1.4) uses anidea of Lesieur-Croisot [76].


Proof of Theorem (1.2). Under the stated conditions we prove that an essentialright ideal E contains a regular element. E is not a nil ideal; it has an elementav ¥= 0 with r(ax) = rfc,2). Either r(aj) r\E = 0 or r(a^) nE^O. In the lattercase, choose a2 e r{a^) n E with a2¥=0 and r(a2) = r(a2

2). If r ^ ) n r(a2) n E # 0,then the process continues.

At the general stage we have a direct sum

atR © ... © ak R © (KaJ n ... n r(afc) n £),

where 0 ^ ^ ) n ... n r(ak-i) n E and afc ^ 0, r(ak) = r(afc2).

The process has to stop, because R has finite rank; let this happen at the k-thstage. Then

r(fli) n ... n r(ak) n E = 0 = r(ax) n ... n r(aft)and hence

r(fl12 + ...+«Jk

2) = (r(fll) n ... n r(«») = 0.

Let Z be the singular ideal of R and zeZ. Then z"JR © r(z") is an essential rightideal for some n > 0 and r(z") is also essential. Hence z"R = 0, Z is a nil ideal andhence is zero. Set c = at

2 + ... + ak2eE; as r(c) = 0 we deduce that cR is essential

by Lemma (1.4). Hence l(c) c Z, so that l(c) = 0 and c is a regular element. Thisestablishes the existence of regular elements in R.

Suppose that a,deR with d regular and set

E = (xejR; axedR).

Then dR is essential and hence so is E, so that E contains a regular element dt. Theright Ore condition is satisfied and R has a right quotient ring Q.

Next suppose that F is an essential right ideal in Q, then F n R is essential injR. Now F n JR has a regular element, which is a unit in Q, and hence F = Q. LetJ be a right ideal and K a right ideal of Q such that J n X = 0 and J © K is essential(use Zorn's lemma); then J © K = Q. Thus the module QQ is semi-simple and Qis a semi-simple ring.

Conversely, let R have a semi-simple right quotient ring Q. Then a right idealE of .R is essential if and only if EQ = Q. To see this, suppose / is a non-zero rightideal of Q, then / n R # 0 and / nRnE # 0, taking £ to be essential in .R. Hence/ n £Q # 0, which means that EQ is essential in Q; then £Q = Q as QQ is a directsum of simple modules. On the other hand, when EQ = Q is given and / is a non-zero right ideal of R then IQ n £Q # 0, trivially and hence / n E # 0, so that £is an essential right ideal of R. These conditions are equivalent to saying that Ehas a regular element, because yeEQ and 1 = ec"1, eei?, ceR and regular. Onthe other hand, when E has a regular element, then EQ = Q and E is essential in R.

We now conclude that JR is a semi-prime ring, because if AT is a nilpotent idealof R, l(N) is essential as a right ideal and has a regular element. Thus N = 0.

Let S = (E/a; aeA) be a direct sum of non-zero right ideals of R which isessential as a right ideal. S has a regular element c, expressible as a finite sum

134 A. W. GOLDIE

Now cR is essential and lies in Jai + .. .+Jan; it follows that A has only the indicesals ...,an, and R has finite right rank.

Finally, the maximum condition holds for right annihilators, because

for any non-empty subset S of R, the subscripts denoting the ring in which theannihilator is taken.

The following corollary is evident.

COROLLARY (1.5). A prime ring R has a simple artinian ring Q as its right quotientring if and only if conditions (2) and (3) hold.

There also follows, since Q is a full ring of matrices over a division ring;Q = Dn, say.

COROLLARY (1.6). A prime ring R with conditions (2) and (3) contains a primering R' which also has Q as its right quotient ring and R' is a full matrix ring Cn, whereC has the division ring D as its right quotient ring.

This corollary is due to C. Faith and Y. Utumi [29].

2. Rings with nilpotent ideals

The earliest result is due to Levitski in 1939 and states that in a right noetherianring JR every nil right ideal is nilpotent. Nowadays this theorem is an easy con-sequence of Lemma (1.3). It readily follows that R has a maximal nilpotent idealN which is um'que, the nilpotent radical and R/N is a semi-prime ring. N is theintersection of the prime ideals of R and indeed is the intersection of the minimalprime ideals, which are finite in number. For further details on radicals seeDivinsky [23] and Jacobson [60].

It is impossible to generalise the procedures of Theorem (1.2) in order to obtainregular elements. Instead the concept has to be taken over factor rings.

Let A be an ideal of R and set<6' (A) = (ceR; cxeA,xeR implies that xe A)l(€ (A) = (CER; xceA, xeR implies that xe A)

<$ (A) = # ' (A) n >(€ (A).

Under the conditions of Theorem (1.2) we have # (0) = W (0) c '#(0). Casesoccur where <#'(0) # '#(0), see [38] and [107].

THEOREM (2.1). Let R be a right noetherian ring with nilpotent radical N and letP l 5 . . . , Pn be the minimal prime ideals of R. Then

(1) «'(0) <= <f(N);

(2)

(3)

(4) let aeR, ce#'(°) then ateR, c^^iN) exist with act = cav


We know that #(JV) = <g'(N) c '<g(N) by applying Theorem (1.2) to the ringR/N. Part (1) of this theorem is due to Djabali [28] and (2), (3) are due to Goldie.(3) is a short way of saying that ce^'(0) and neN together imply that c + ne<tf'(O).This shows that no modification of the proof of Theorem (1.2) will give regularelements. Part (4) is implicit in Small [108]. It is tempting to call this the pseudoOre condition. It can be used to prove (1). This approach leads to the valuabletheorem due to Small [108].

THEOREM (2.2). A right noetherian ring R has a right quotient ring Q, which is aright artinian ring, if and only if

That #(0) = #(N) is necessary had been proved earlier by Talintyre [111].

After this point the quotient problem for noetherian rings becomes very difficult

and no decisive results have been obtained. There are three aspects to the problem:

(i) when does a noetherian ring R have regular elements;

(ii) when are there enough regular elements to satisfy the right Ore condition;

(iii) what is the structure of a quotient ring ?

For commutative noetherian rings the matter is settled by the maximal primesof zero, these are the maximal annihilator ideals and they are finite in number. Anelement is regular if and only if it does not lie in any of these primes, such elementsexist if and only if the ring is faithful (Rx = 0 implies that JC = 0). The secondquestion is trivial and, as for the third, a quotient ring is a semi-local ring in which theJacobson radical has non-zero annihilator.

Rings with zero singular ideal are of some interest. Such a ring, if commutative,is semi-prime but in general the structure is very complicated. They are not knownin the artinian case except when indecomposable right ideals are uniserial. Theyare then determined as direct sums of blocked triangular matrix rings over divisionrings. See Goldie [40], Gordon [45] and Colby-Rutter [19]. The assumption thati? has zero singular ideal is useful technically, because ce# ' (0) implies that cR isan essential right ideal and hence l(c) = 0. Thus #(0) = #'(0). Djabali [28] settlessome cases of the quotient problem for rings with zero singular ideal.

In the general case when the existence of the quotient ring is assumed, relatedproperties can be studied.

THEOREM (2.3). Let R be a right noetherian ring with a right quotient ring Qand N be the nilpotent radical of R. Then NQ is the nilpotent radical of Q. Let P bea prime ideal of R then either PQ = Q or #(0) c #(P) and PQ is a prime idealof Q with PQnR = P.

Let P' be a prime ideal of Q, then P' n R is a prime ideal of Rand (P' nR)Q = P'.

This theorem is well known for commutative rings but its generalization is notimmediate. It depends on Theorem (2.1) and so far is only known for noetherianrings. There is no published proof at present.

136 A. W. GOLDIE

Some necessary conditions for the existence of the quotient ring are obtainedas follows. The transfer ideal of #(0) is the largest ideal T such that

c + te #(0) for all c e #(0), t e T.

There is also a transfer right ideal T", defined in the same way. Clearly they areuniquely defined and T <= T". In order to fix the idea we remark that the Jacobsonradical of a ring is the transfer ideal of the group of units.

Suppose that a noetherian ring R has a right quotient ring Q and let T, T" be thetransfer ideals of R. It is a consequence of Theorem (2.3) that

JnR=T = T',

where J is the Jacobson radical of Q. It follows that T is a semi-prime ideal ofR. This may, of course, be true in a ring which does not have a quotient ring but atpresent the best result known is that T 3 N, see Theorem (2.1).

We conclude our remarks on the quotient problem with a reference to Small[107] which gives an example which is a right and left noetherian ring but does nothave a quotient ring on either side. This ring satisfies a polynomial identity.

For relief we next turn to an easy case, the principal ideal rings. A ring R withunit element is a pri-ring when its right ideals are principal (single generator). Thefollowing theorems are due to Goldie [39]. See also Procesi [96] and Johnson [68]for a different treatment.

THEOREM (2.3). (1) A semi-prime pri-ring is a finite direct sum of prime pri-r ings.(2) A prime pri-ring is isomorphic to a full ring of matrices over a right Ore domain.

THEOREM (2.4). A pri-ring which is left noetherian is the direct sum of a semi-prime pri-ring and an artinian pri-ring. It has a right quotient ring which is an artinianpri-ring.

Difficulties arise with pri-rings which are not left noetherian, see the examplein Section 1 and also Jategaonkar [62].

A number of other problems have been studied from time to time. An old00

conjecture due to Jacobson enquires whether f] J" = 0, where J is the Jacobsonn = l

radical, holds in a right noetherian ring. A counter-example is given in Herstein[52], following an idea of Lance Small. The problem is still open for right and leftnoetherian rings.

An ideal T of a ring R is a primary ideal if AB a T, where A, B are ideals of R,implies that either i c T or Bk c T, together with the corresponding property whenA and B are interchanged. A strongly-primary ideal T is a primary ideal such thatR/T has an artinian quotient ring. Is a primary ideal always strongly-primary?This seems unlikely and a counter-example should throw light on the nature ofthe quotient process.

Many difficulties stem from the lack of a representation of an ideal as an inter-section of primary ideals. It is fashionable to avoid this property in commutativework, but we cannot indulge this whim. Indeed primary decomposition does not


hold in algebras, for if it held in group algebras then all finite groups would be soluble.The only decomposition theory which has anything to offer in the general case

is the tertiary theory. There have been many attempts to deal with the questions ofuniqueness and existence, most theories succeed in one place and fail in the other.The tertiary theory succeeds in both respects. Unfortunately, it is very difficult torelate to other aspects of the ring structure. We refer the reader to Curtis [20], forearlier work and Lesieur-Croisot [75] for the tertiary theory. Here we restrict oursummary to the case of a right noetherian ring R having unit element, which hassome simplifying features.

Let M be a right K-module. Index its essential submodules as Ma (a e A) anddefine the radical of M to be

rad M = (xeR; Ma x = 0, some a e A).

Clearly rad M is an ideal. When M is finitely generated, rad M is a finite intersectionof prime ideals.

Let / be a right ideal then the tertiary radical of / is

r(/) = rad(K-J)

when / is an ideal of R we have r(I) 3 I, but this need not hold for right ideals.Let Rbea simple artinian ring, / = eR, e2 = e ^ 0, 1, then r(I) = 0. The definitioncan be rephrased for left modules and left ideals and indeed for bimodules. Thusfor an ideal T we can define a radical R(T) by the rule:

R(T) is the set of elements aeR such that every ideal B c£ T contains an idealB' $ T with B'a c T.

This definition can be rephrased on the left as well. It follows that an ideal Thas four tertiary radicals; as a right or left ideal and on the right or left as a (two-sided) ideal. These can differ, certainly R(T) ZD r(T), but equality occurs forartinian rings. This is a nice exercise in socle play. Whether R(T) = r(T) fornoetherian rings is not yet settled. In order that the reader can keep his bearings, wemention that in a commutative ring these radicals all reduce to

VT = (xeR; x"eT for some n > 0).

A right ideal / is tertiary when

bRa d , b$I, implies that aer(I).

Then r(I) is a prime ideal. A right ideal V is irreducible when R — V is a uniformright .R-module. Then V is a tertiary right ideal. It follows that every proper rightideal is a finite intersection of tertiary right ideals.

THEOREM (2.5). Let I be a proper right ideal of the right noetherian ring R andI = It n ... n Ik where the Ij are tertiary right ideals with associated prime idealsPjlj = l,...,k. Then

The intersection of a finite set of tertiary right ideals, each having the sameassociated prime ideal P, is again P-tertiary. This enables a decomposition to be

138 A. W. G0LD1E

brought into reduced or normal form (the associated primes are distinct), as in thecommutative theory. A uniqueness theorem now follows.

THEOREM (2.6). Let I = Iln...nIh = Jln...r\Jk, where the decompositions

are reduced, the Ia, Jp being tertiary right ideals. Then h = k and the two sets of

associated prime ideals coincide.

These results can be carried out for left ideals and repeated for two-sided idealsT, using R(T) instead of r(T).

In the formal sense the theory is entirely satisfactory and indeed is the onlypossible theory for which these theorems hold, see [101]. Nevertheless, it has proveddifficult to apply to the study of the structure of noetherian rings, because the natureof a tertiary ideal is difficult to understand. Moreover the tertiary radical has todestroy the partial order even for two-sided ideals. For example, r(P) = P for allprime ideals P of R. Since (0) c P, preservation of partial order would imply thatr(0) c= P for all primes P, which would mean that r(0) is the nilpotent radical.This is certainly not the case as it would lead to the existence of a primary decom-position for ideals.

3. Localisation

Non-commutative local rings occur surprisingly often, although it is difficultto provide a natural construction of a local ring in the general situation and itsvalue as a method is as yet uncertain in non-commutative rings. A local ring isdefined in Cartan-Eilenberg [15] to be a ring R with unit element such that theelements which do not have a left inverse form a left ideal M. Then M is a two-sidedideal which contains all proper left and right ideals; in particular, R/M is a divisionring. In practice the definition is found to be too restrictive and is modified asfollows. A local ring is a ring R with unit element such that the Jacobson radicalJ is a maximal ideal and R/J is an artinian ring.

Evidently J is the only maximal ideal of R. A full ring of n x n matrices over aCartan-Eilenberg local ring is a local ring in the new sense, although it is unlikelythat all local rings have this form.

A more general construction is studied in Gabriel [33], see also Bourbaki [14]and Murdoch [89], proceeding in a context much more general than is consideredhere and in effect using partial endomorphisms. These do not lead, as far as can beseen, to local rings of our sort except in special cases. All methods reduce to theusual construction for commutative rings.

A set 3F of right ideals of a ring R is topological provided that

(1) & is a filter,

(2) aeR, Fe^ implies that

a'^F = (xeR; axeF)e^.

R is thus regarded as a topological ring having a basis of neighbourhoods of zeroconsisting of right ideals.


When M is a right .R-module, define the ^-singular submodule to be

= (meM; r(m) e &),

where r(m) = (xeR; mx = 0).A closure operation is defined on submodules N of M by

cl^ N = (m e M; mF cz N, some F e &).

Topological sets 3F, $ of right ideals can be multiplied together, the setcomprises those right ideals / for which cl^/ belongs to 2P'. The set tpy is againtopological and contains the ideals FG\Fe&r, Ge&.

A topological set $F is idempotent when $F2F = !F'. This is the important caseand occurs when J5" satisfies the additional axiom:

(3) c l , / e & if and only if / e 3F.

The useful construction of an idempotent topological set is obtained by takinga multiplicatively closed subset S of elements of R which does not contain zero anddefining 3F(S) to be the set of right ideals F of R such that a~lF meets S for everyaeR. It is readily verified that ^(S) satisfies axioms (1), (2), (3).

A multiplicatively closed subset S cz R which does not contain zero, satisfiesthe right Ore condition when, for given aeR, seS there exist a^ e R , si eS withasx = sav Then the set of right ideals which contain a right ideal sR, seR, is anidempotent topological set and is a subset of #"(S). When S has only regular elements,the right Ore condition is essentially that considered in Section 1 and is equivalentto the existence of a classical right quotient ring Rs, with elements of the formas'1, aeR, seS.

It is now assumed that we are dealing with a (left and right) noetherian ring Rwith unit element. Let P be any prime ideal of R; P ^ R.

We know from Theorem (1.2) that <€ = >cti, where

# = W(P) = {ceR; cxeP implies that xeP}l(€ = '<#(P) = {ceR; xceP implies that xeP}

This depends on knowing that the prime ring R/P has a quotient ring on left andright. # is multiplicatively closed. Moreover cR + Pe^ffi, whenever c e # .

Now set J5" = &{$>) and define a corresponding topological set of left ideals<g = <$(<$). For example Rc+Pe<&.

Now define a closure operation for any two-sided ideal A or R by

k(A) = (xeR; GxF a A, some Ge&, Fe&).

k(A) is an ideal containing A and k(kA) = kA.This operation enables symbolic powers of P to be defined; there are two sorts;

left symbolic powers Hn and right symbolic powers Kn where n = 1, 2, . . . given by

P; Hn+l = k(PHn); Kn+i = k(KnP).

140 A. W. GOLDIE

The following theorems were given in Goldie [41].

THEOREM (3.1). Let R be a right and left noetherian ring, P a prime ideal andHn, Kn defined as above. Then

(1) HrHs c Hn, whenever r+s = n, n = 1, 2 , . . . ,

(2) R/Hn satisfies the left and right Ore conditions with respect to the set#„ = {c+Hn\ ceW} and #„ is a set of regular elements,

(3) Hn = Kn,

(4) <6n is the set of all regular elements of R/Hn,

(5) Hn is a P-primary ideal.

Properties (2) and (4) show that R/Hn has a right and left quotient ring whichwe denote by Q(ri).

THEOREM (3.2). Under the conditions of Theorem (3.1),

(1) the quotient ring Q(ri) of R/Hn is an artinian primary ring,

(2) there is a canonical epimorphism Q(n + \)~* Q(n),

(3) the inverse limit Q, of the rings Q(n),n = 1,2,... under the canonicalepimorphisms is a complete local ring,

(4) Q is a full kxk matrix ring over a complete local ring L,

(5) L is a local ring in the sense of Cartan-Eilenberg.

Q is complete in the topological sense, the topology being that arising from thepowers of the maximal ideal $ of Q; this corresponds to the Zariski topology for

00

commutative rings. The topology is hausdorff, because the property f] $" = 0n = l

holds. The integer k of part (4) is obtained from the quotient ring Q(l) of the primering R/P. Q(l) is isomorphic to the full matrix ring Dk, D a division ring. Notethat g/i& ^ Dk.

Since the classical (or commutative) method of localisation is not possible ingeneral the " usual" local ring has to be sought in another way. An existencetheorem is the following.

THEOREM (3.3). Let {(^l a e A} be the family of local rings with correspondingmaximal ideals {Ma; a e A} such that

Ra&czQ; Pac=Macijfr; aeA.Then for all aeA

( l ) M a = jfrnQa,(2) {R,V}<^Qa,(3)Hn = Ma

nnR, « = 1,2,...,(4) QJMa s g/jft.Let Q = (f)Qa; aeA) and M = ((]Ma; aeA). Q is a local ring with maximal

ideal M. Q is a member of the family.


The local ring Q can be derived by construction as follows.

THEOREM (3.4). Define a sequence of rings Q(n), n = 1,2,..., where

RczQ' czQ"c . . .c(2( f l )c . . . e g

with Q' = {R,^-1} and Q(n+1) = { Q ^ , ^ " 1 } ; M(n) = til n Q(n); and ^(ll)

m rte n«# Q(fl). Set Q* = UQ(n)> ^ M* = UM(n), « = 1,2,.... 7%«i Q* = Qand M* = M, as defined in Theorem (3.3).

The ring {R, # - 1 } is the subring of (5 generated by R and the inverses ofelements of (€. However it should be noticed that in these theorems the symbol Rhas been used to denote R/H, where H = f]Hn, n = 1,2, — It is R/H which is em-bedded in (5. This accords with the commutative case, since there we have

0P(n), n = 1, 2, ... = (xeR; xc = 0 some ce<tf(P)).

The usual procedure for clearing out zero divisors before localising has to be avoided,because (xeR; xc = 0, some ce^(P)) is no longer an ideal. See McConnell [83]for an interesting example.

The classical construction occurs in a number of cases and its non-commutativeform is as follows. To localise on the right it is necessary to suppose that the elementsof #(P) satisfy the right Ore condition.

For each pair a e R, ce #(P) we need the existence of a pair at eR, c1 6 ^(P)with acy = cat. The ^-component of zero is

(0 :#) = (xeR; xc = 0 some ce#).

Then (0 : #) is a two-sided ideal of R. Moreover if the elements of %> are left regular(xc = 0, x = 0) then they are regular. It follows that on passing to the ring

R — —, . = R, the image # of # is a set of regular elements which satisfy the right(U: <r) ^

Ore condition. Form the right quotient ring of JR with respect to #; this ring is alocal ring RP. The procedure works whenever R is right noetherian and one finds thatRP is also right noetherian. The restriction imposed by the Ore condition is on theprime ideal P rather than on R itself because act = cat has only to be verified whenae P. See Goldie [41] for details or an account of a similar type in Gabriel [33].

It is quite likely that in a given ring R, classical and non-classical localisationsmay both exist, the particular sort dependent on whether we have a " good " or" bad " prime ideal to deal with. The serious weakness of the general constructionlies in that it is not known whether Q or Q are noetherian or otherwise.

Many other problems arise with local rings. Is every factor ring of a local ring,even when noetherian, also a local ring? This is a question of whether two-sidedideals are closed in the Zariski topology and can be asked for one-sided ideals also.A discussion of the relationship between these and classical localisation is given in[41].

142 A. W. GOLDIE

This question is generalised in McConnell [84] and Hinohari [57]. Let R be a ringwith a two-sided ideal / such that f]I" = 0, n = 1,2,..., and suppose that

(i) R/I is a right artinian ring,

(ii) / is finitely generated as a right ideal,

(iii) R is complete in the /-topology.

For a ring with these properties we have

THEOREM (3.5). Let E be a hausdorjf right R-module ((]EIn, (n = 1,2,...) = 0)and F a submodule of E. Consider the conditions:—

(i) F is a finitely generated R-module,

(ii) For each k > 0, then exists n > 0 with F n El" <= FIk,

(iii) F is a closed submodule of E, viz F = f](F+EIn); n = 1,2,....

Then (i) implies (ii) and (iii). / / E is finitely generated then (ii) implies (i) and theinduced topology on F coincides with the l-topology.

The implication (i) -»(iii) is due to Hinohari; the remainder to McConnell.

4. Universal enveloping algebras

Suppose that L is a Lie algebra of finite dimension over a field k and let U(L)be the universal enveloping algebra of L. Refer to Section 1 and Jacobson [61;Chapter 5]. We give here a brief description of properties of U(L) which are relatedto the ideas given so far. This is but a small part of the extensive literature on Liealgebras and is restricted to the cases of nilpotent or solvable algebras.

We need to define the algebras An = Alt(k), n— 1,2,.... These are given inHirsch [58] and D. E. Littlewood [81]. An is the algebra generated over k by 2«generators phqx where / = 1,...,« subject to the equations

[Pi, qi\ = Pi qi-qiPi = i; \j>i, qj\ = [Pi,pj\ = fo,> qjl = o (/ # ; ) .

The elements p^'q^' •••pninqn

Jn (ia,j* > 0) are a basis of the vector space An overk. The centre of An is isomorphic to k and

An = Ax®...®A1.

An is a left and right noetherian domain; it is a simple algebra when k hascharacteristic zero.

THEOREM (4.1). Let L be a nilpotent Lie algebra of finite dimension over a fieldk of zero characteristic. The following properties of a two-sided ideal IofU = U(L)are equivalent:—

(1) the centre of U/I is afield,

(2) U/I is isomorphic to An for some n,

(3) / is a maximal ideal of U,

(4) / is a primitive ideal of U.


This theorem is due to Dixmier [26] when k is algebraically closed and non-denumerable. He also proved most of the statements in the general case except for(4) => (3) which is due to Quillen [100] and (3) => (2) which is due to Gabriel andNouaze" [34]. The latter paper introduces some new methods of a functorial nature.

THEOREM (4.2). Under the circumstances of Theorem (4.1), the following prop-erties of an ideal IofU are equivalent,

(1) I is a prime ideal of U,

(2) U/I is an integral domain,

(3) the centre of R/l is an integral domain.

A prime ideal / with property (2) is said to be completely prime. In the case ofnon-zero characteristic there is a counter-example due to McConnell.

Let L be the nilpotent Lie algebra over a field k of characteristic 2, with a basisx, y, z such that

[x, y] = z and [x,z] = [y,z] = 0.

There is a chain of prime ideals in U,

Pl = (0)aP2 = (x2)(zP3 = (x,y,z).

Also P l 5 P3 are completely prime but R/P2 has a quotient ring which is a 2 x 2simple matrix algebra. It is interesting to note that localisation is classical at each P,-.

Part of these theorems still holds for solvable Lie algebras and is again due toDixmier [27].

THEOREM (4.3). Let L be a solvable Lie algebra over a non-denumerable,algebraically closed, field k of zero characteristic. Then every prime ideal P of U iscompletely prime. Also P is a primitive ideal if and only if the centre of the quotientrint of U/P is k.

THEOREM (4.4). With conditions on the field k as in Theorem (4.3) and L anyLie algebra of finite dimension, the following properties are equivalent

(1) L is solvable,

(2) every primitive ideal of U is completely prime,

(3) every prime ideal of U is completely prime.

Let R be any ring with 1 and / a two-sided ideal of R. A set xu ...,xneR is a

centralising set of generators of / when / = (xu ...,xn) and

n

x, e centre R; Xt+Cx,, ...,X;_,)e centre- - ; i > 1.

The property can be generalised by using the subset of R,

JV(K) = (xeR; xReRx).

A set Xj,..., xn E R is a normalising set of generators of the ideal / when I = (xlt ...,xn)

144 A. W. GOLDIE

and / nCieN(R);

\xi> •'•>xi-i

These definitions and the following theorem are due to McConnell [82], [83].

THEOREM (4.5).

(1) Let L be a nilpotent Lie algebra over any field. Then any two-sided ideal ofU(L) has a centralising set of generators.

(2) Let L be a solvable Lie algebra over an algebraically closed field of zerocharacteristic. Then any two-sided ideal of U(L) has a normalising set ofgenerators.

When this theorem is applied to the ideal M of U(L), which is generated by L,then a theorem due to Kaplansky [70], or its extension due to Fields [32], showsthat the global dimension of U is ^ dim L. It follows that these dimensions areequal, by a theorem due to Roy [105].

This theorem has relevance to the question of localisation in U(L). Refer toSection 3 and to McConnell [83].

THEOREM (4.6) Let R be a right and left noetherian ring with unit element andP be a prime ideal of R. If either

(i) every ideal of R has a centralising set of generators,

or (ii) P is completely prime, H = 0 and every ideal of R has a normalising set ofgenerators, then

Hn = (xeR; xceP") = (x; cxeP"), some ce<$(P).

COROLLARY Let R and P satisfy one of the conditions (i), (ii). / / P has the ARproperty then Rp is classical. This is true for a nilpotent Lie algebra L.

An ideal / of R is said to satisfy the AR-condition when for each right ideal Eof R there exists n > 0 such that

EnFczEI.

The following theorem along these lines appears in Nouaz6-Gabriel [34].

THEOREM (4.7). Let R be a right noetherian ring with unit element and I be anideal of R having a centralising set of generators. Let E be a finitely generated R-module, F a submodule of E. Then for each k > 0, these exists n > 0 such that

FnEFc FIk.

Theorems on chains of primes are available in U(L), these are due to Gabrieland Rentschler [35].

THEOREM (4.8). Let L be any Lie algebra of dimension n. Then any chain ofprimes in U(L) including (0) but excluding R, has length ^ n + l.


5. Rings with polynomial identities and related rings

The centroid C{R) of a ring R is the ring of those endomorphisms co of theadditive group R+ with the property

w(xy) = (cox)y = x(coy), x, yeR.

Let {xj} be a set of non-commuting indeterminates and consider a polynomial

p[x]=p[xl, ...,*„] = ScoO")*,-, ...*,-k,

which has coefficients co(i) in C(R) and (/) = (/l5..., ik), the summation being overall ilt..., ik e 1,2,...,«. The polynomial /?[*] is a polynomial identity for i? if/*[ri» •••>rn] = 0 f° r a ^ ri» •••5r,,e# and co(i)R+ ^ 0 for at least one (/).

For brevity we say that R is a Pl-rmg. The theory was initiated by Kaplanskyin [69] where the important Theorem (5.1) is proved. The early development isgiven in Jacobson [60], much of its direction is towards the solution of the Kuroschproblem for Pi-algebras over a field.

For Theorem (5.1) we follow Amitsur [6].

THEOREM (5.1). Let R be a right or left primitive ring satisfying a non-trivialidentity p[x] = 0 of minimal degree d. Then R is a central simple algebra of dimensionn2 over its centre and d = In.

Conversely, a central simple algebra of dimension n2 over its centre satisfies apolynomial identity of minimal degree In.

The identity satisfied in Theorem (5.1) will be unique to a scalar multiple(from C(R)) and is the standard identity

with summation over all a = (iu ..., i2n) belonging to the symmetric group on(l, . . . ,2n).

We are concerned here with the effect of the following theorem due to Posner[94], which caused a decisive shift in the theory of Pi-rings.

THEOREM (5.2). Let R be a prime ring which has a polynomial identity of minimaldegree d. Then R has a right and left quotient ring Q which is a central simple algebraof dimension n2 over its centre and d = In. Let C be the centre of Q, then RC = Q.Moreover Q satisfies the same identities as R.

The theorem was proved by Posner for prime algebras and extended in Amitsur[6] to the general case. Amitsur made use of the theory of ultra-products, but thiscan be avoided without affecting his clear approach to the theorem. See Goldie [44].The following lemma is fundamental.

LEMMA (5.3). Let R be a prime ring with a polynomial identity and c be a regularelement. Then cR ZD Rd where d is a regular element.

This lemma shows that R satisfies a strong form of the right Ore condition sinceact = ca^ holds with cx = d independent of the choice of the element a. No chain

BULL. 2 2

146 A. W. GOLDIE

conditions on right or left ideals are assumed in Theorem (5.3), because the PI-hypothesis guarantees them. See Posner [94].

The extension of these results to semi-prime rings with PI needs care, since apolynomial identity p[x] may become trivial, thus all coefficients are zero, afterpassage to a factor ring. In particular, a prime ideal P of R is trivial with respectto p[x] when coR c P for all its coefficients co. The following theorem from [6]copes with this difficulty.

THEOREM (5.4). A semi-prime ring R satisfies an identity p[x] = 0 of degree d if andonly if R is a subdirect sum of two semi-prime rings Rl}R2 such that p[x] is trivialon R2 and Rt £ Hn, where n < [d/2] and H is a commutative semi-prime ring.

One would expect Rt to have a quotient ring, but this is unsettled, except underadditional conditions, which are such that Rt has a semi-simple quotient ring whichis a semi-simple algebra of finite dimension over its centre. This had been provedearlier in Small [108].

Posner's theorem provides a foundation for a non-commutative affine geometrydeveloped in papers by Amitsur and Procesi [7, 97, 98]. In [7] a form of the Hilbertnullstallensatz is proved for Pi-factor rings of the free algebra J F ^ , ..., xn], F a field,xu ...,xn non-commuting indeterminates. One deduces that finitely generatedalgebras over F which have a polynomial identity are Jacobson rings. This latterproperty asserts that every prime ideal is an intersection of maximal ideals. Theresult is extended in the following theorem; see [7].

THEOREM (5.5). Let Q be a commutatuve Jacobson ring and R = O[A-19 ..., xn]be a finitely generated Q-ring in the centre of R, which satisfies a polynomial identity.Then

(1) R is a Jacobson ring,

(2) if M is a maximal ideal of R then QnM is a maximal ideal of Cl and R/Mis a finite-dimensional algebra over the field Cl/Cl n M,

(3) if Q is an F-Hilbert algebra then so is R.

An algebra A over F is an .F-Hilbert algebra if every simple factor algebra isfinite-dimensional over F. The identities which hold in R are assumed to have theircoefficients in Q, and to be non-trivial. This theorem has been improved by Procesi[98] with the aid of an interesting form of ring extension.

Let R be a subring of a ring S and set

CS(R) = (x e Sx xr = rx, all x e R)

the centraliser of R in S. Then S is an extension of R when S = RCS(R). S is afinitely generated extension of R when S = R[xlt ...,*„] with x,eCs(i?). Noticethat the xt need not commute among themselves.


THEOREM (5.6). / / S is a finitely generated extension of a Jacobson ring R andhas a polynomial identity, then

(1) S is a Jacobson ring,

(2) if M is a maximal ideal of S, then RnM is a maximal ideal in R and S/Mhas finite length over R/R n M,

(3) if R is an F-Hilbert algebra, then so is S.

Such extensions of noetherian rings are not noetherian; see [98] for a counter-example. Procesi does prove the following results.

THEOREM (5.7) A finitely generated Fl-algebra over a commutative noetherianring satisfies the ascending chain condition for semi-prime ideals.

THEOREM (5.8). Let S be a finitely generated extension of a ring R and assumethat S satisfies a polynomial identity with coefficients ± 1. Then

(1) if R has ascending chain condition on prime ideals then so has S,

(2) if R has descending chain condition on prime ideals then so has S,

(3) if R has finite rank on primes then so has S.

The rank of R on primes is the maximum length of a chain of prime ideals of R.A restricted type of finitely generated algebra is studied in Small [108] and earlier

in Curtis [21]. Let A be a commutative noetherian right with unit. An 4-algebraR with unit is finitely generated if it is finitely generated as an i?-module. In thesecircumstances R is a right and left noetherian ring, as also is its centre. Such algebrasinclude group algebras of finite groups over A and the ring of endorphisms of afinitely generated module over A. Small proves that finitely generated algebras canbe embedded in artinian rings and later gives a counter-example to show that thisdoes not hold for general noetherian rings. There are also theorems on chains ofprime ideals which anticipate the generalisations of Theorem (5.8). He also obtainsa successful solution of the Jacobson conjecture, namely (f) J"; n = 1,...) = 0,where J is the Jacobson radical of R. This is not true for one-sided noetherian ringsas an example in Herstein [52] shows, but finitely generated algebras are left andright noetherian rings and the question is still open for these.

The study of nil and nilpotent rings is part of the area of polynomial identities.There is an interesting case in Higman [56] which deals with rings which are nil ofbounded index. In Herstein and Small [54] the relationship between nil and nil-potent rings is dealt with in the following result.

THEOREM (5.9). A nil ring with ascending chain condition on left and rightannihilators is nilpotent. The same conclusion holds for a nil ring which has thecondition on right annihilators and has finite rank {see Section 1).

A counter-example due to Sasiada shows that the ascending chain condition onright annihilators is not enough. However it is enough when the ring has a poly-

148 A. W. GOLDIE

nomial identity. In Kaplansky [69] we find the earlier result that any nil PI ring islocally nilpotent.

6. Simple and related ringsIn this section we give results obtained by a closer study of rings which have

simple artinian rings as their quotient rings. One underlying problem is to determinethe structure and representation of simple noetherian rings. Some special cases havebeen dealt with in earlier sections. Thus Kaplansky's theorem shows that a simplering with polynomial identity is artinian and Theorem (2.3) shows that a simpleprincipal right ideal ring is a full matrix ring over a simple right noetherian domain.But the general problem is difficult, and infuriatingly so, because results for simplerings have a tendency to be true for some wider class of rings. However this fieldof study has another powerful motivation, the development of a non-commutativearithmetic. This has a history which goes back to the quaternion integers and whichreceived an early development due to the work of Artin [8], Asano [9] and Chevalley[17].

We begin with some results about simple rings. It is known that a simple rightnoetherian ring R which has Q as its right quotient ring has a unit element whenQ has infinite centre. See Robson [103]. The following results are due to Hart [49].

THEOREM (6.1). A simple ring R with unit, which has a uniform right ideal, isisomorphic to eKn e, where K is a right Ore domain and e is an idempotent element ofthe matrix ring kn for some positive integer n.

Unfortunately, there is nothing invariant here, e, n, K are all variables and wecannot even assert that K is isomorphic to End^ (U, U) for a uniform right idealU. Note that a simple ring with unit has a uniform right ideal if and only if it hasfinite right rank. Hart also proved that EndR(U, U), U a uniform right ideal, is asimple ring if and only if U is projective as an fl-module. This leads to the followingimprovement of Theorem (6.1) in a special case.

THEOREM (6.2) Let R be a simple ring with unit, which has the ascending chainconditions on right annihilators, and which possesses a projective uniform right ideal.Then R is right noetherian if and only if R is isomorphic to eKn e, where K is a simpleright noetherian domain and e is an idempotent in Kn.

In order to study arithmetical properties a number of concepts are needed whichgeneralise those of the commutative theory. These are given in Jacobson [59] andRobson [104], and we mention only those which are needed for the statement ofresults.

A right order R in a quotient ring Q is a sub-ring of Q such that every element ofQ has the form ac~x\ a,ceR with c regular. Right orders R, S in Q are equivalentif there exist units c, d, e,f in Q with

cRdczS and eSfczR.

A right order R in Q is a maximal (equivalent) order if no right order which properlycontains R is equivalent to it.


A right order R is bounded if any right ideal cR, where c is a regular element,contains a non-zero two-sided ideal. For example, a prime ring with polynomialidentity is a bounded order because of Amitsur's lemma.

A simple ring with unit is a maximal equivalent order. However the class of ordersequivalent to a given one need not have a maximal member in the general situation.

A right order R with unit is an Asano right order if the (fractional) i?-ideals in Qform a group under multiplication. The following theorem is due to Robson [104]and had been proved earlier by Asano [9] for bounded orders.

THEOREM (6.3). Let R be a right order with unit element in a quotient ring Q.The following properties are equivalent:

(1) R is an Asano right order,

(2) JR is a maximal right order and every integral R-ideal is a projective rightR-ideal,

(3) For each integral R-ideal T there exists an R-ideal T * with

TT* = T*T = R,

(4) The R-ideals form an abelian group under multiplication.

In order to develop an arithmetic of right ft-ideals it is necessary to involve theorders which are equivalent to R and to strengthen the assumptions on R. For thispurpose Robson [104] introduced the (right) Dedekind right order which is a maximalright order in which every integral right .R-ideal is projective. This also generalisesan earlier idea due to Asano who had studied such orders in the bounded case. Thering of polynomials F[x, y] over a field F of characteristic zero such that xy—yx = 1is a noetherian hereditary domain and is simple. It is a Dedekind right (and left)order but is not a bounded order.

THEOREM (6.4). Let R be a Dedekind right order. The class of all right S-ideals,where S runs over all right orders equivalent to R, forms a Brandt groupoid underordinary multiplication, wherever this is defined.

The definition and some properties of Brandt groupoids are to be found inJacobson [59].

Thus far the quotient ring Q can be arbitrary but representations of Asano andDedekind orders can be found when Q is a simple artinian ring; these parallel theTheorems (6.1) and (6.2) and are proved by the same methods as in Hart [49].

THEOREM (6.5). Let R be an Asano right order in a simple artinian ring. ThenR is isomorphic to eKn e for some idempotent e in Kn, where K is a right order in adivision right D, and n > 0.

When R has a projective uniform right ideal then K is an Asano right order in D.For Dedekind right orders this leads to a characterisation, because they are

right hereditary rings.

150 A. W. GOLDIE

COROLLARY. A right Dedekind prime ring has the form eKn e, where e is idem-potent in Kn, and K is a right Dedekind domain. Conversely, if K is a right Dedekinddomain, then eKn e is a right Dedekind prime ring.

These rings have also been studied by Michler [88], who shows that localisationsat prime ideals are classical for bounded Asano prime orders and deduces that theserings are Dedekind right orders. Robson (unpublished) has also shown that for anon-simple right Dedekind prime ring R every proper factor ring is an artinianprincipal ideal ring (as in the commutative case) and if the Jacobson radical of Ris non-zero then R itself is a (non-artinian) principal ideal ring. Once again thecase of simple rings evades analysis for the present.

The theory of maximal orders in a different sense has also been studied byAuslander and Goldman [11]; this follows a line of development in simple algebras;see Deuring [22]. Here an order over a noetherian domain D, is a subring R of acentral simple algebra A over the quotient field K of D such that R is a finitelygenerated D-module and RK = A. An order R is maximal if R is not properlycontained in any other order in A. These orders are always right and left orders inthe Asano sense mentioned earlier and they are bounded, because A satisfies astandard polynomial identity, being a finite dimensional simple algebra. The maintheorem obtained is as follows.

THEOREM (6.6). Let D be a discrete rank one valuation ring and R be a maximalorder in a central simple algebra A over the quotient field K of D. Then

(1) all maximal orders in A are conjugate to R,

(2) R is a principal ideal ring,

(3) R is a full matrix algebra over a maximal order in a division algebra.

A localisation argument then shows that a maximal order R over a dedekindring D (commutative) is the ring of endomorphisms of a finitely generated projectivemodule over a maximal order in a division algebra. Thus the theory turns around andreturns to Theorem (6.2).

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152 A. W. GOLDIE

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154 SOME ASPECTS OF RING THEORY

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325-341.

University of British Columbiaand Leeds University.

some aspects of ring theory

Documents

van der waerden

ascending chain condition

infinite direct sums

finitely generated extension

full matrix ring

ascending chain conditions

finitely generated algebras

central simple algebra