the krull–gabriel dimension of a serial ring
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The Krull–Gabriel Dimension of a Serial RingGena Puninski a ba Department of Mathematics , The Ohio State University at Lima , Lima, Ohio, USAb Department of Mathematics , The Ohio State University at Lima , Galvin Hall 4240, Lima,OH, 45804, USAPublished online: 01 Feb 2007.
To cite this article: Gena Puninski (2003) The Krull–Gabriel Dimension of a Serial Ring, Communications in Algebra, 31:12,5977-5993, DOI: 10.1081/AGB-120024862
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The Krull–Gabriel Dimension of a Serial Ring
Gena Puninski*
Department of Mathematics, The Ohio State University at Lima,Lima, Ohio, USA
ABSTRACT
We prove that every serial ring R has the isolation property: everyisolated point in any theory of modules over R is isolated by a mini-mal pair. Using this we calculate the Krull–Gabriel dimension of themodule category over serial rings. For instance, we show that thisdimension cannot be equal to 1.
Key Words: Serial ring; Krull–Gabriel dimension; Isolationproperty.
1991 Mathematics Subject Classification: 16L30; 16B70.
*Correspondence: Gena Puninski, Department of Mathematics, The OhioState University at Lima, Galvin Hall 4240, Lima, OH 45804, USA; E-mail:[email protected].
COMMUNICATIONS IN ALGEBRA�
Vol. 31, No. 12, pp. 5977–5993, 2003
5977
DOI: 10.1081/AGB-120024862 0092-7872 (Print); 1532-4125 (Online)
Copyright # 2003 by Marcel Dekker, Inc. www.dekker.com
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1. INTRODUCTION
Let A be a finite dimensional algebra over a field k, and let mod(A) bethe category of finite dimensional A-modules. An object of many investi-gations in the representation theory is the category mod(mod(A)) offinitely presented additive covariant functors from mod(A) to the cate-gory of k-vector spaces. For instance, the Krull–Gabriel dimension of thiscategory, KG(A), is of special interest (see Schroer, 2000).
It was noticed by Burke (2001) that over a general ring R this notionsplits into two parts: we may consider the Gabriel dimension of R, andits finitely presented variant, the Krull–Gabriel dimension ofR. The choicedepends on whether we factor out arbitrary simple functors or just thosesimple functors which are finitely presented in the relevant factor category.
These dimensions may be different, but, by Burke (2001, Thm. 5.1),they coexist.
In this paper we consider the Krull–Gabriel dimension, KG, of aserial ring. In particular we prove that for a serial ring R, KG(R) existsiff the lattice of right (left) ideals of R has Krull dimension (in the senseof Gabriel and Rentschler). Moreover, if the Krull dimension of R isequal to a, then KG(R) does not exceed a� a, and we show that it isprecisely a� a for some classes of serial rings. Here a� a stands for theCantor’s sum of a and a.
We also prove that the Krull–Gabriel dimension of a serial ring cannot be equal to 1. This result seems to be similar to the theorem of Krause(1998) that for any finite dimensional algebra over an algebraically closedfield, its Krull–Gabriel dimension is not equal to 1. Herzog (1991) provedthat this is true for any artin algebra.
All the results of this paper are tightly connected with the calculationof the Cantor–Bendixson rank of the Ziegler spectrum over a serial ringR, CB(ZgR), made by Puninski (1999) and Reynders (1999). In fact wewill prove that the so-called isolation property holds true over any serialring R: every isolated point in the Ziegler spectrum of any theory ofR-modules is isolated by a minimal pair.
An immediate consequence of this result is that the following inva-riants of a serial ring R are equal: (1) the Krull–Gabriel dimension ofR; (2) the Cantor–Bendixson rank of the Ziegler spectrum over R; (3)the m-dimension of the lattice of all positive-primitive formulae over R.Having proved this result, instead of calculating the Krull–Gabrieldimension directly, we use the arithmetic of positive-primitive formulaeover a serial ring developed by Puniski (to appear).
For instance we prove that if R is a semi-duo serial ring of (finite)Krull dimension n, then the Krull–Gabriel dimension of R is equal to 2n.
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2. PRELIMINARIES
Let mod(R) be the category of finitely presented (right) modules overa ring R. By F(R) we denote the category of covariant additive functorsfrom mod(R) to the category of abelian groups, and let mod(modR) bethe full subcategory of F(R) consisting of finitely presented functors. Itis well known that mod(modR) is an abelian category. As in Burke(2001) (see also Krause, 2001) we define the Krull–Gabriel filtration ofmod(modR) as a non-decreasing sequence of Serre subcategories (theoriginal definition of Geigle (1985) uses contravariant functors).
Take F�1¼f0g and recursively define Fa such that Faþ1 consists offunctors that are of finite length in the quotient category mod(modR)=Fa
(and Fl¼S
m<lFm at limit stages). If at some stage a the localizing cate-gory generated by Fa is equal to F(R), and a is minimal such, we say thatthe Krull–Gabriel dimension of R is equal to a.
For more of this, in particular how the Krull–Gabriel dimension isconnected with the Gabriel dimension, the reader is referred to Burke(2001). For instance, KG(Z)¼ 2, but the Gabriel dimension of Z isequal to 1. Note that our terminology is the same as in Krause (2001) orPrest (2000).
The basic notions of the model theory of modules, including theZiegler spectrum, may be found in Prest’s book (1987). For instance,ppM (m) will denote the pp-type of an element m of a module M, i.e.the set of all pp-formulae which are satisfied by m in M.
Let j, c be pp-formulae over a ring R. We will use (j=c) to denote,depending on the context, (1) the interval [j^c, j] in the lattice of allpp-formulae over R; (2) the pair of pp-formulae (j, c) (usually withc < j), or (3) the basic open set (j=c) in the Ziegler spectrum ZgR overR.
Let T be a theory of modules over a ring R. A pair of pp-formulae(j, c) is called minimal, if the interval (j=c) is simple in T. We saythat T has the isolation property, if for any extension T 0 of T, every iso-lated point in ZgT0 is isolated by a minimal pair.
Recall that a module M is called distributive, if the lattice of submo-dules of M is distributive. We say that M is endo-distributive, if M is dis-tributive as a module over its endomorphism ring S¼End(M ). Similarlywe say that a module M is pp-distributive, if the lattice of pp-definablesubgroups of M is distributive.
A module M is said to be uniserial, if the lattice of submodules of Mis a chain, and M is serial, if M is a direct sum of uniserial modules. Forinstance, every uniserial module is distributive.
A ring R (with a unit 1) is serial, if RR is a serial right R-module and
RR is a serial left R-module. Thus R is serial iff there exists a collection
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e1, . . . , en of orthogonal idempotents such that (1) 1¼ e1þ � � � þ en;(2) every right module eiR is uniserial, and every left module Rei is uni-serial. Usually we consider a serial ring R with a fixed system of orthogo-nal (indecomposable) idempotents e1, . . . , en.
If R is a serial ring, then every ‘diagonal’ ring Ri¼ eiRei is uniserial,and eiRej is an Ri-Rj-bimodule. Let us define a non-increasing sequenceJac(a) of two-sided ideals of R. Put Jac(0)¼ Jac(R), and Jac(aþ 1)¼T
n Jacn(a) at non-limit stages. If l is limit then set Jac(l)¼T
m<l Jac(m).By Muller (see Puninski, 2001, Prop. 1.30) the Krull dimension of a
serial ring R, Kdim(R), is equal to a iff a is the least ordinal such that theideal Jac(a) is a nilpotent. In particular, the right Krull dimension of Ris equal to the left Krull dimension of R.
3. THE ISOLATION PROPERTY
Theorem 3.1. Let T be a theory of modules, and suppose that every inde-composable pure-injective module M in ZgT is contained in a basic open set(j=c), such that the interval (j=c) in the lattice of pp-definable subgroupsof M has width. then the isolation property holds for T.
Proof. Since the above condition is inherited by extensions of T, it suf-fices to prove that every isolated point in ZgT is isolated by a minimalpair.
LetM be an isolated point of ZgT. By hypothesis,M is contained in abasic open set (j=c) (with c < j), such that the lattice of pp-definablesubgroups of M between c(M ) and j(M ) has width. Then this latticecontains a subinterval that is a chain, so we may assume that the interval[c(M ), c(M )] is a chain from very beginning.
By Ziegler (1984, 4.9) a basis of neighborhoods for M in ZgT can bechosen from the collection of subpairs of the pair (j=c). Thus we mayassume that (j=c) isolates M in ZgT.
Now we prove that the lattice of pp-formulae in T between c and jis a chain. Otherwise there are incomparable pp-formulae y1, y2 betweenc and j in T.
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So both pairs (y1=y2) and (y2=y1) are nontrivial in T. Since they are sub-pairs of (j=c), they both must isolate M. But, by the choice of (j=c),either y1(M )� y2(M ) or y2(M )� y1(M ) holds, hence at least one of thesepairs is closed on M, a contradiction.
Thus the lattice of pp-subgroups in T between c and j is a chain. ByPrest (1987, Th. 10.2) the pair (j=c) generates on every super-decompo-sable pure-injective module in T. From the remark in Prest (1987, p. 213)we conclude that M is isolated by a minimal pair in ZgT. &
Proposition 3.2. Let T be any theory of modules over a serial ring R. Thenthe isolation property holds for T.
Proof. Let M be an indecomposable pure-injective R-module. Choose0 6¼m2M, hence mei 6¼ 0 for some i. Then M is in the basic open set(ei j x=x¼ 0), where ei jx denotes the pp-formula ‘‘ei divides x’’. But, byPuninski (2001, L. 11.4), the lattice of pp-definable subgroups ofM belowMei is a chain.
Thus it remains to apply Theorem 3.1. &
Note that (see Puninski, 2001, Ch. 12) there is a serial ring with asuper-decomposable pure-injective module. Therefore Proposition 3.2applies beyond the case when the continuous part of T is zero.
The theory T is called distributive if the lattice of all pp-formulae ofT is distributive.
The following is a characterization of these theories.
Proposition 3.3. For a theory T (over an any ring) the following areequivalent:
(1) T is distributive.(2) Everypure-injectivemodelofT isendo-distributive (pp-distributive).(3) Every pure-injective indecomposable module in T is endo-uniserial
(pp-uniserial).
Proof. (1)) (2). Let M be a pure-injective model of T, S¼End(M ).We prove that M is a distributive left S-module. By Tuganbaev (1998,1.15 (iii)) it suffices to check the distributivity on cyclic S-submodulesSm, m2M. Since M is pure-injective, every such submodule is of theform p(M ), where p¼ ppM (m).
So let m, n, k2M, p¼ ppM (m), q¼ ppM (n) and r¼ ppM (k). It isenough to check the inclusion
pðMÞ \ ðqðMÞ þ rðMÞÞ � ð pðMÞ \ qðMÞÞ þ ð pðMÞ \ rðMÞÞ:
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Let l2 p(M )\ (q(M )þ r(M )), i.e. l2 (j^ (cþ y))(M ) for anyj2 p,c2 q,and y2 r. Since T is distributive, it follows that l2 ((j^c)þ (j^ y))(M ).But M is pure-injective. Then Prest (1987, Cor. 2.3) yields l2 ( p(M )\q(M ))þ ( p(M )\ r(M )).
Since every endo-distributive module has a distributive lattice ofpp-subgroups, M is pp-distributive.
(2)) (1) is clear, because the lattice of pp-formulae of T coin-cides with the lattice of pp-subgroups of a ‘‘large’’ pure-injectivemodel of T.
(2)) (3). Let M be a pure-injective indecomposable model of T,S¼End(M ). By hypothesis SM is distributive. Since S is a local ring,by Stephenson (see Tuganbaev, 1998, Th. 2.1) SM is a uniserial module.
(3)) (2). Let M be a pure-injective model of T. By Prest (1987,Cor. 3.8) M is a direct summand of a direct product of indecom-posable pure-injective models of T. But by Tuganbaev (1998, 8.6 (i))endo-distributivity preserves under direct products and directsummands. &
Proposition 3.4. The isolation property holds for every distributive theory.
Proof. By Proposition 3.3 and Theorem 3.1. &
The standard example of a distributive theory is the theory of allmodules over a commutative Prufer ring. Thus we obtain the following.
Corollary 3.5. The isolation property holds for every theory of modulesover a commutative Prufer ring.
Since we are mainly working with serial rings, let us derive some stan-dard conclusions only in this case.
Proposition 3.6. Let R be a serial ring. Then the following invariants of Rare equal:
(1) The Cantor–Bendixson rank of the Ziegler spectrum of R.(2) The m-dimension of the lattice of all pp-formulae over R.(3) The Krull–Gabriel dimension of R.
Proof. By Proposition 3.2, the isolation property holds for everytheory of modules over a serial ring R. Thus (1) and (2) are equivalent
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by Prest (1987, Prop. 10.19). The remaining equivalence is proved inBurke (2001, Thm. 4.5). &
4. m-DIMENSION
If a, b are elements of a latticeL, (a=b) will denote the interval [a^ b, b].Let L be a lattice with 0 6¼ 1. We define a non-decreasing sequence of
congruences �a on L. Let ��1 be the trivial congruence. If �a has alreadybeen defined, put a�aþ1 b if the interval (a=b) has finite length in thelattice La¼L=�a. At a limit stage l we set �l¼
Sm<l�m.
The m-dimension of L, mdim(L), is equal to a, if a is the least ordinalsuch that La consists of one element. For instance, mdim(L)¼ 0 iff L isfinite, and mdim(L) is not defined iff L contains the ordered set of therationals Q as a sublattice.
We say that the interval (a=b) is a-simple, if mdim(a=b)¼ a and forevery c2 (a=b), either mdim(a=c) < a or mdim(c=b) < a holds. Thenmdim(a=b)¼ a iff (a=b) can be decomposed into finitely many a-simpleintervals.
Lemma 4.1. Let L be a lattice with 0 6¼ 1 and let a2L. Then the m-dimen-sion of L is equal to the maximum of the m-dimensions of the intervals (a=0)and (1=a).
Proof. Let mdim(L)¼ a, mdim(a=0)¼ b, and mdim(1=a)¼ g.From (a=0), (1=a)�L it follows that b, g� a, hence max(b, g)� a.Let us prove that a�max(b, g). Note that, if L0 is a convex sublattice
of L, then any congruence �d defined on L0 is the restriction of the con-gruence �d defined on L. In particular this applies to the intervals (a=0)and (1=a).
We may assume that b� g. From mdim(a=0)¼ b it follows that0�b a in (a, 0), hence a�b 0 in L. Then a�g 0 in L, and similarly a�g 1in L. Thus 0¼ 1 in L=�g, hence mdim(L)� g. &
Let L1 and L2 be chains with 0 6¼ 1. By L¼L1�L2 we will denote themodular lattice freely generated by L1 and L2 with respect to the relations01¼ 02 and 11¼ 12 (i.e. the smallest and the largest elements of L1 andL2 are identified). For instance, if L2 consists of two elements, thenL1�L2¼L1.
It is well known (see Gratzer, 1978, Thm. 13) that this lattice isdistributive. Moreover it is quite easy to describe the shape of elements
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of this lattice. Let us represent elements a2L1 and b2L2 by rectangles(in the plane L1�L2) in the following way:
Then L is isomorphic to the lattice of subsets of the plane L1�L2 gene-rated by these rectangles with respect to usual set theoretic operations \and [. For instance the resulting figures for elements aþ b and a^ b arethe following:
Thus every element of L can be (uniquely) represented as (an^ b1)þ(an�1^ b2)þ � � � þ (a1^ bn), where a1 < a2 < � � � < an2L1 and b1 <b2 < � � � < bn2L2, i.e., as a descending ladder with just finitely many steps(some of the steps may be infinite). For instance, if a1 < a2 < a32L1 andb1 < b2 < b32L2, then (a3^ b1)þ (a2^ b2)þ (a1^ b3) looks as follows:
Given a < a0 2L1 and b < b0 2L2, the interval (a0 ^ b0=aþ b) can bethought as the rectangle P¼ (a0=a)� (b0=b):
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If a, b are ordinals, then a� b will denote their Cantor sum. To calculatethis, just represent a¼oa1n1þ � � � þoaknk and b¼oa1m1þ � � � þoakmk,where a1 > a2 > � � � > ak (some of ni, mi may be zero). Thena� b¼oa1(n1þm1)þ � � � þoak(nkþmk). In particular a� b¼ b� a.
Proposition 4.2. Let L1 and L2 be chains with 0 6¼ 1, and L¼L1�L2.Then mdim(L)¼mdim(L1)�mdim(L2).
Proof. Let a < a0 2L1, b < b0 2L2, and let P be the rectangle(a0=a)� (b0=b), i.e., the interval (a0 ^ b0=aþ b) in L. By induction on gwe prove that P is g-simple in L iff (a0=a) is a-simple in L1, (b
0=b) is b-sim-ple in L2, and g¼ a� b.
As a basis of the induction take g¼ 0. If (a0=a) is simple (i.e.,0-simple) in L1 and (b0=b) is simple in L2 then by the above descri-ption P is simple in L. Suppose that P is simple in L. If (a0=a) isnot simple in L1 then there exists a12L1 such that a < a1 < a0.Clearly (a0 ^ b)þ (a1^ b0) is between a0 ^ b0 and a0 ^ b0 ^ (aþ b)¼(a^ b0)þ (a0 ^ b), a contradiction.
Thus we may assume that we have already established the descrip-tion of d-simple rectangles P in L for every d < g. Let P¼(a0=a)� (b0=b) be such that mdim(a0=a)¼ a0, mdim(b0=b)¼ b0, anda0 � b0 < g. Then P is decomposed into finitely many rectangles whosesides are d-simple for certain d < a� b. By Lemma 4.1 we obtainmdim(P)¼ a0 � b0.
Let us assume that (a0=a) is a-simple in L1, (b0=b) is b-simple in L2,
where a� b¼ g. We prove that P is g¼ a� b-simple (it is not g0-simplefor every g0 < a� b by the induction hypothesis). Otherwise P can bedecomposed as P1[P2 such that both figures P1 and P2 have m-dimen-sion not less than g.
The case when P is sectioned only by one horizontal or one verticalline is clear. Suppose that the board of P1 and P2 is a nontrivial ladder as
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shown below. We use an additional induction on the number of steps ofthis ladder.
Suppose that mdim(b0=b1) < b. Then the m-dimension of the rectangleP 0 ¼ (a 0=a)� (b 0=b1) is less than g. Also the interval (b1=b) is b-simple. Thuswemay removeP 0 fromP1 andP2 without changing them-dimension. Sincewe have decreased the number of steps, the result will follow by induction.
Otherwise mdim(b0=b1)¼ b. Since the interval (b0=b) is b-simple, itfollows that mdim(b1=b) < b. Suppose that mdim(a0=a1) < a. Then them-dimension of the rectangle (a0=a1)� (b0=b) is less than g by induction,hence mdim(P1) is less than g (since P1 is contained in this rectangle), acontradiction.
Thus we may assume that mdim(a0=a1)¼ a. Since (a0=a) is a-simple,it follows that mdim(a1=a) < a. Then (by induction) both rectangles(a1=a)� (b0=b) and (a0=a1)� (b1=b) have m-dimension less than g. SinceP2 is contained in the union of these rectangles, we conclude thatmdim(P2) < g, a contradiction.
So it remains to prove that there are no other simple rectangles Pwith dimension a� b. First suppose that P¼ (a0=a)� (b0=b),mdim(a0=a)¼ a 0, mdim(b0=b)¼ b0 and a0 � b0 > g¼ a� b. By symmetrywe may assume that there is g0 < a0 such that g0 � b0 g. Sincemdim(a0=a) > g0, this interval is not g0-simple. Therefore there existsa12 (a0=a) such that mdim(a1=a) g0 and mdim(a0=a1) g0. Let us splitP¼P1[P2 in the following way
Then, by induction, mdim(P1) g and mdim(P2) g, hence P is notg-simple.
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It remains to consider the case when a0 � b0 ¼ a� b but one of seg-ments, say (a0=a) is not a0-simple. Then there is a12 (a0=a) such thatmdim(a1=a)¼mdim(a0=a1)¼ a0. Decomposing P as above we obtain thatP is not g-simple.
As the final step we deduce that the m-dimension of the rectangle(11=01)� (12=02) is equal to mdim(L1)�mdim(L2). But this rectangle isthe interval (11^ 12=01þ 02)¼ (1=0)¼L. &
By L1�0 L2 let us denote the modular lattice freely generated by L1
and L2 (without the identification of 0 and 1).
Corollary 4.3. Let L1, L2 be chains with 0 6¼ 1 and L¼L1�0 L2. Thenmdim(L)¼mdim(L1)�mdim(L2).
Proof. It is clear that the lattice L1�L2 is a factor lattice of L, thereforemdim(L)mdim(L1�L2)¼mdim(L1)�mdim(L2).
For the converse let L10 be obtained from L1 by adding a new smal-
lest element 0�1 and a new largest element 1þ1 , and similarly for L2. Since0 6¼ 1 in L1 and L2 we have mdim(L1)¼mdim(L1
0) and mdim(L2)¼ 3mdim(L2
0). If L0 ¼L10 �L2
0 then mdim(L0)¼mdim(L10)�mdim(L2
0)¼ 3mdim(L1)�mdim(L2). But clearly L1�0 L2 is a sublattice of L0. &
5. KRULL–GABRIEL DIMENSION
In this section we investigate the Krull–Gabriel dimension of a serialring.
Recall that the Krull dimension of a lattice L, Kdim(L), is defined asfollows. Kdim(L)¼ 0 iff L has the descending chain condition, i.e. if L isartinian. By induction on ordinals we define Kdim(L) aþ 1, if there is adescending chain a1 > a2 > � � �, ai2L such KG(aiþ1=ai) a for each i.For a limit ordinal g we set Kdim(L) g if Kdim(L) b for every b < g.
Now the Krull dimension of L is the smallest ordinal a such thatKdim(L) aþ 1 fails.
In general the Krull dimension and the m-dimension of a lattice Lmay differ drastically (although they coexist). For example, if L¼oaþ 1,then Kdim(L)¼ 0 but mdim(L)¼ a. But for a serial ring they are almostthe same.
Recall that an idempotent e of a ring R is said to be indecomposable,if the projective right module eR is indecomposable. If R is a serial ring,then e is indecomposable iff eR is a uniserial module.
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Lemma 5.1. Let e be an indecomposable idempotent of a serial ring R.Then the following invariants are equal:
(1) The Krull dimension of the lattice of all submodules of eR.(2) The Krull dimension of the lattice of cyclic submodules of eR.(3) The m-dimension of the lattice of cyclic submodules of eR.
Proof. Let L be the lattice of all submodules of eR, and let L0 be thelattice of all cyclic submodules of eR. Since L0 is a sublattice of L,Kdim(L0)�Kdim(L).
Let us prove that Kdim(L0)Kdim(L), hence Kdim(L0)¼Kdim(L).If Kdim(L) a, then there exists an order inverting embedding f :oa!L.If b2oa, then f(b) f(bþ 1) yields that rb2 f(b) n f(bþ 1) for some rb2R.Thus f(b)� rbR f(bþ 1)� rbþ1R implies rbR rbþ1R. Therefore themap b! rbR defines an order inverting embedding g from oa to L0. ThusKdim(L0) a.
Let us extend g to oaþ 1 by sending 1 to the zero ideal. Sincemdim(oaþ 1)¼ a we obtain mdim(L0) a.
It remains to prove that mdim(L0)�Kdim(L). If mdim(L0) does notexist then there is a countable dense subset in L0, hence Kdim(L) does notexist. So we may assume that Kdim(eR) is defined.
By Puninski (2001, Prop. 1.30) Kdim(eR) is the least a such thateJac(a)n¼ 0 for some n. By induction on b� a we prove that every inter-val (rR=rsR) for r2 eR, s2 Jac(b)k has m-dimension not more than b.Decomposing s in a product we may assume that s2 Jac(b) n Jac(b)2. Ifmdim(rR=rsR) > b then there is a principal ideal I� eR such thatrsR� I� rR and mdim(I=rsR) b, mdim(rR=I ) b. Clearly I¼ rtRwhere s¼ ts0 for some t, s0 2R. Since s 62 Jac(b)2 by symmetry we mayassume that t 62 Jac(g)l for some l and g < b. Then, by induction,mdim(rR=rtR)� g, a contradiction. &
The following lemma gives a neccesary condition for the existence ofthe Krull–Gabriel dimension of a serial ring.
Lemma 5.2. Let R be a serial ring. If the Krull–Gabriel dimension of Rexists then the Krull dimension of R is defined.
Proof. Suppose that the (left) Krull dimension of R is undefined. Thenfor some i the left module Rei does not have Krull dimension. Thereforeby Lemma 5.1 there exists a dense chain of principal submodules of Rei.
Thus there are rq2Rei, q2Q such that rq2Rrq0 iff q q0. Let jq,q2Q be divisibility formulae rq j x. Clearly jq!jq0 iff q q0, hence this
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chain of pp-formulae is dense. Thus the m-dimension of the lattice of allpp-formulae over R does not exist. By Proposition 3.6 the Krull–Gabrieldimension of R does not exist. &
So in further considerations we restrict ourselves on the case of serialrings with Krull dimension.
Now we find an upper bound for Krull–Gabriel dimension of a serialring. This result is very similar to considerations in Reynders (1999, Cor.3.27) concerning Cantor–Bendixson rank (there is a small lapse in thatpaper which is originated in the author’s paper, Puninski (1999)—insteadof 2a there should be a� a everywhere).
Proposition 5.3. Let R be a serial ring with Krull dimension a. Then theKrull–Gabriel dimension over R is less than or equal to a� a.
Proof. Let L be the lattice of all pp-formulae over R. By Proposition 3.6it suffices to prove that mdim(L)� a� a.
By Puninski (2001, L. 11.1) every interval (e1 j x=x¼ 0) in L is gener-ated by two chains: L1 consisting of divisibility formulae a j x, a2Re1,and L2 consisting of annihilator formulae xb¼ 0, b2 e1R. Since L1 isanti-isomorphic to the lattice of cyclic submodules of Rei, mdim(L1)� a.Also L2 is isomorphic to the lattice of cyclic submodules of e1R, whichgives mdim(L2)� a.
Then mdim(e1 j x=x¼ 0)� a� a by Proposition 4.2. By Lemma 4.1it remains to prove that mdim(x¼ x=e1 j x)� a� a. But this interval isisomorphic to the interval (e j x=x¼ 0) where e¼ e2þ � � � þ en, so wecan complete the proof by induction. &
Corollary 5.4. Let R be a serial ring. Then the Krull–Gabriel dimension ofR exists iff the Krull dimension of R is defined.
Proof. By Lemma 5.2 and Proposition 5.3. &
Note that the above-mentioned bound for the Krull–Gabriel dimen-sion of a serial ring is optimal. For instance (see Puninski, 1999, Cor. 3.6),if V is a commutative valuation domain of Krull dimension a, then theCB-rank of ZgV is equal to a� a. Hence the Krull–Gabriel dimensionof V is equal to a� a.
Now we obtain a lower bound for the Krull–Gabriel dimension of aserial ring.
Let e be an indecomposable idempotent of a serial ring R.For a, b2Re put a�l b if b2Ra (i.e. Rb�Ra), and a<l b if Rb�Ra.
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Similarly for c, d2 eR we set c�r d if d2 cR, and c <r d if dR� cR.Finally for e, f2R we write e� f if f2ReR, and e < f if RfR�ReR. Notethat <l and <r are linear orders whereas < is just a partial order.
Let L be a linear order with 0 6¼ 1, and let e be an indecomposableidempotent of a serial ring R. We say that the functions f :L!Re,g :L! eR form a boundary pair, if for every a, b2L, a < b we havef (a) <l f (b), g(a) <r g(b) and f (a)g(a) < f (b)g(b).
Proposition 5.5. Let R be a serial ring, and let f :L!Re, g :L! eR bea boundary pair. If the m-dimension of L is equal to a, then the Krull–Gabrieldimension of R is not less than a� a.
Proof. By Corollary 4.3 (and Proposition 3.6) it suffices to embed L�0 Linto the lattice P of all pp-formulae over R.
Note that for l, l 0 2L we have f(l) j x! f(l 0) j x iff l 0 � l, andxg(l)¼ 0! xg(l 0)¼ 0 iff l� l 0. Thus there are two copies of L inside P:one is given by divisibility formulae jl (x)¼ ‘‘f(l ) j x’’, and the other isgiven by annihilator formulae cl (x)¼ ‘‘xf(l)¼ 0’’. Both these chainshave m-dimension a.
It remains to prove that the lattice of pp-formulae generated by thesetwo chains is freely generated. Otherwise for some l < l 0 2L we havef (l ) j x^ xg(l 0)¼ 0! f (l 0) j xþ xg(l)¼ 0. By Puniski (to appear, L. 3.1)it follows that f (l 0)g(l 0)� f (l)g(l), a contradiction. &
It is known by Krause (1998) and Herzog (1991) that there exists nofinite dimensional algebra A such that the Krull–Gabriel dimension of Ais 1. The corresponding question for Zg(A)¼ 1 is still open. In the follow-ing corollary we consider this property for serial rings.
Corollary 5.6. Let R be a serial ring. Then each of the following invariantsis not equal to 1: (1) the Krull–Gabriel dimension of R; (2) the Cantor–Bendixson rank of the Ziegler spectrum of R; (3) the m-dimension of thelattice of all pp-formulae over R.
Proof. We may assume that R has Krull dimension. If R is artinian thenit has a finite representation type, hence all these dimensions are zero.
So we may assume that R is not artinian. Then (see Puninski, 2001,L. 1.32) there exists an indecomposable idempotent e such that the uni-serial ring eRe is not artinian. Thus we may assume that R is a uniserialnon-artinian ring with Krull dimension.
Since R has Krull dimension, there exists p2R such that Jac(R)¼pR¼Rp.
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By Proposition 5.5 it suffices to construct a boundary pair f,g :oþ 1!R. Let us define f(n)¼ g(n)¼ pn for n2o and f(1)¼ 0. Ifm < n2o then clearly f(m)¼ pm <l p
n¼ f(n) and similarly for g. Alsof(m)g(m)¼ p2m and f(n)g(n)¼ p2n. Clearly the assumption p2m2Rp2nR¼p2nR leads to a contradiction. &
Now it is easy to calculate the Krull–Gabriel dimension for someclasses of serial rings.
Corollary 5.7. Let R be a serial ring of Krull dimension 1. Then the Krull–Gabriel dimension of R is equal to 2.
Proof. By Corollary 5.6, the Krull–Gabriel dimension of R is greaterthan 1. It remains to apply Proposition 5.3. &
According to Puninski (2001, Ch. 7), every right noetherian (non-artinian) serial ring R has Krull dimension 1. Thus the Krull–Gabrieldimension of R is equal to 2. Note that Generalov (1997) provedthat the right Gabriel dimension of a right noetherian serial ring is equalto 1.
A serial ring R is called semi-duo, if for every r2 eiRej we have eithereiRr� rR or rRej�Rr. By Puninski (2001, Thm. 2.13) a serial ring Ris semi-duo iff every finitely presented indecomposable (left or right)R-module has a local endomorphism ring.
Proposition 5.8. Let R be a uniserial semi-duo ring of Krull dimension a.Then the Krull–Gabriel dimension of R is equal to a� a.
Proof. By Proposition 5.3 we obtain KG(R)� a� a.Let R0 ¼R=Jac(a). By Puniski (to appear, Cor. 9.4) R0 is a semi-duo
uniserial domain of Krull dimension a. Then by Reynders (1999, Cor.3.27) the Cantor–Bendixson rank of ZgR0 is equal to a� a. It remainsto apply Proposition 3.6. &
For a serial semi-duo ring we have the following weaker form of theprevious proposition.
Corollary 5.9. Let R be a serial semi-duo ring of finite Krull dimension n.Then the Krull–Gabriel dimension of R is equal to 2n.
Proof. Every ‘‘diagonal’’ ring Ri¼ eiRei is a uniserial semi-duo ring.Moreover, by Puninski (2001, L. 1.32), there exists i such that the Krull
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dimension of Ri is equal to n. By Proposition 5.8 the Krull–Gabrieldimension of Ri is equal to 2n. Then KG(R) 2n.
It remains to apply Proposition 5.3. &
Conjecture 5.10. Let R be a serial ring of finite Krull dimension. Then theKrull–Gabriel dimension of R is even.
It is clear from Puniski (to appear) that the structure of the lattice oftwo-sided ideals of a serial ring should play a crucial role in calculationsof Krull–Gabriel dimension.
Remark 5.11. Let R be a serial ring. Then the lattice of two-sided idealsof R is distributive.
Proof. If I is a two-sided ideal of R, I¼Li, j eiIej (as an abelian group).
It remains to note that eiRej is uniserial as a right ejRej-module (or lefteiRei-module). &
ACKNOWLEDGMENT
This paper was written during the visit of the author to the Univer-sity of Manchester supported by EPSRC grant GR=R44942=01. Hewould like to thank the University for the kind hospitality.
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Received July 2002Revised April 2003
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