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Endomorphism rings and tensor productsof linearly compact modulesCarl Faith a & Dolors Herbera ba Department of Mathematics , Rutgers University , NJ08903, USAb Departament de Matemàtiques , Universitat Autònoma deBarcelona , Bellaterra (Barcelona), 08193, Spain E-mail:Published online: 27 Jun 2007.

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COMMUNICATIONS IN ALGEBRA, 25(4), 12 15-1255 ( 1 997)

ENDOMORPHISM RINGS AND

TENSOR PRODUCTS OF

LINEARLY COMPACT MODULES

Department of Mathematics Rutgers University

New Brunswick, NJ08903, USA

Departament de Matematiques Universitat Autbnoma de Barcelona 08193 Bellaterra (Barcelona), Spain

email: dolorsQmat .uab.es

The term linearly compact (= LC.) refers to 1.c. in the discrete topology:

n/r is a 1.c. R-module if each system of congruences

indexed by a set I and where the Mi are submodules of M, has a solu-

tion x whenever it has a solution for every finite subsystem, equivalently, if

n i E F ( x , + Mi) # 0 holds for every finite subset F C I, then i t holds for F = I. Let R be a commutative ring, M an R-module, and let A :- EndR M denote HomR(M, PI). If A is LC. as an R-module we say A is R-l.c., and we study the following question:

&&(hl): If R is a 1.c. commutative ring and M is a 1.c. R-module, is the

endomorphism ring A of M R-l.c.?

Copyright O 1997 by Marcel Dekker, Inc

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1216 FAITH A N D HERBERA

Any commutative 1.c. ring R has a XIorita duality induced b ~ . HomR( : E) , xvhere E is the minimal injective cogenerator over R and then the category

of reflexive modules is exactly the category of linearly compact R-modules

(see 5 1.3). We study whether this category is closed by the Hom and tensor functors. More precisely, let M * = HornR(ib!, E ) be the E-dual of :Lf. Then

.If* is 1.c.: and one asks:

&Q(lZf, N) : If &I and N are R-1.c.. is HomR(M, N) R-l.c.?

arid similarly:

&Q(:\d @ N) : Is M @ R !V 1.c. over R?

Since .-l r (M iZ/I')' as R-modules; and more general. since

as R-modules, then the universal truths of all these questions are equivalent (cf.Theorem 1 . l 4 ) , and is denoted by &RC(R) , or just ERC , and called the

Endomorphism Ring Conjecture for R. It is clear that the tensor product of finitely generated modules is finitely

generated, so if we are working over a linearly compact ring, it will be linearly

compact. The solution seems to be not so easy for tensor products of finitely

embedded linearly compact modules. In fact if the tensor product of finitely

embedded linearly compact R-modules is linearly compact then R will satisfy

ERC (Theorem 1.14).

We verify ERC for maximal valuation rings (Theorem 3.3), complete Koe-

therian rings (Theorem 6.2) and linearly compact rings constructed from rings satisfying E R C , that is: for linearly compact algebras over a ring sat-

isfying IRC (Proposition 1.15) and a lexicographic extension of a ring satis-

fying ERC (Theorem 4.12). These classes of rings include the main sources

of examples of commutative linearly compact rings.

We further verify & Q ( M ) for the following kinds of 1.c. R-modules M.

(1) M finitely generated (= f.g.) (Theorem 1.10).

(2) M finitely embedded (= f.e.) (Theorem 1.12).

(3) hf almost finitely generated (= a.f.g.), that is .'lf is not f.g, but every

proper submodule of 31 is f.g. (Proposition 6.4, the proof requires

M'eakley's theorems [\V] ) .

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LINEARLY COMPACT MODULES

(4) M is a rational extension of a f.g. submodule -1-, equivalently HornR(ibliAV, .LI) = 0 (Theorem 1.13).

(5) Any torsion-free module it1 over a reduced ring R (Theorem 2.3).

We study also whether the endomorphism ring of a linearly compact mod- ule .\I over a non necessarily linearly compact commutative ring IZ is a lin-

early compact ring. Our approach to the problem in closely related to ERC:

(a) first we look for a linearly compact con~mutative ring S such that

EndR(iV1) = Ends(:W),

(b) then if S satisfies ERC, or 1L1 is a kind of S-module such thzt EQ(ib1) is satisfied, we can conclude that EndR(,U) is linearly com?act.

h'e show that this method works for 1.c. modules over noethe.:ian rings

(Theorem 6 . 2 ) , finitely embedded 1.c. modules (Corollary 5.7) and uniserial

1.c. modules (Theorem 5.10). The idea of associating a linearly compact ring to a linearly compact

module is inspired in V6mos' paper [Vl] and generalizes the work done for

artinian modules, cf. the papers by Facchini [Fa], Ballet [Ba, page 3751, Sharp

[Sh] and Camps and Facchini [C-F, Theorem 2.1 and 2.21.

We do not know whether step (a) is true in general. We can a l ~ a : ~ . s assume

that the ring is local (Lemma 1.3), and we choose as a candidate for S the completion of R in the M-topology. We prove that this completion coincides

with the completion of R in the trE(M)-topology (where t r E ( M ) denotes the trace of M on the minimal injective cogenerator E) and with E n d ~ ( t r ~ ( i \ / l ) ) (Lemma 5.4 and Proposition 5.5). We show that S is a linearly compact ring if and only if t rE(M) is R-1.c. (Proposition 5.5).

One of the advantages of taking this kind of completions is that they preserve also the lattice of submodules of the tensor product (Lemma 1.4).

Hence we can also deduce that the tensor product of linearly compact mod-

ules over a noetherian ring is linearly compact (Theorem 6.2), that the tensor

product of artinian modules is a module with finite length (this is a conse- quence of Theorem 6.2, but we give an elementary proof in Proposition 6.1) and that the tensor product of uniserial linearly compact modules is uniserial and linearly compact (Theorem 5.10).

The paper is divided into six sections. The first one introduces most of

the basic tools for the rest of the paper and tries to make it as self-contained

as possible. Sections 2, 3 and 4 are devoted to study the ERC question and in section 5 we work with the problem of associating a linearly cornpact ring

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12 18 FAITH A N D HERRERA

to a linearly compact module. In section 6 we study in detail the noether-

ian situation, it gathers the main ideas in the paper to give some structure

theorems for linearly compact modules over noetherian rings: the results on linearly compact rings with linearly compact quotient field of section 3, on in-

decomposable linearly compact modules of section 4 and on i'd-completion~

of section 5. However we want to remark that some of the results in this

section were already obtained by Zoschinger in [Zo]. ours is just a different

approach.

Throughout R ~vill denote a commutative ring (unless otherwise is stated)

with 1, and modules will be unital. If ikf is an R-module we denote by

EndR(i\l) its endomorphism ring (or simply by End(.\I) when the ring R is

clear). If S C M , annR(S) = ( r E RlSr = 0) . J ( M ) denotes the Jacobson

radical of the module M and Soc(M) its socle.

Let M and N be R-modules recall that the trace o j !\il on !V is the sub-

module of N!

1.1 Fini teness p rope r t i e s of l inearly c o m p a c t modules . \Ire summa-

rize here some of the module theoretical properties of linearly compact mod-

ules. We will use them in the rest of the paper sometimes with no previous

acknowledgement.

The category of linearly compact right R-modules (R can be a non com-

mutative ring) form a Serre subcategory of mod-R. That is if M is a right

R module and N is a submodule of M, M is linearly compact if and only if

N and M/IV are linearly compact (c.f [X, Chapter 1.31).

A linearly compact module M has finite Goldie dimension (c.f [X, Chapter 1.3]), thus any quotient of M has finite Goldie dimension. Zoschinger showed

in [Zo] that over a commutative noetherian ring a 1.c. module M is finitely

generated by artinian, that is there exist a finitely generated submodule iV

of 22.1 such that M I S is artinian. Zijschinger proof is very involved with the

structure of commutative noetherian rings. Independently Enochs proved in

a elementary way, that a linearly compact module over a complete noetherian ring is finitely generated by artinian (c.f. [En, Proposition 1.31). A look at

Enochs proof shows that he is only using that the ring is noetherian and that

all the quotients of the module have finite Goldie dimension. Enochs' idea

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LINEARLY COMPACT MODULES 1219

can be generalized to arbitrary rings and gives the following result that was

coinniunicated to us by Peter Vbmos.

1.1 Propos i t i on . (A . McGuire and P. Vgmos) Let R be a r ing (con.am,utative

o r n o t ) and :ZfR a n o n zero right R -modu le such that a n y quotient of :LI has finite Goldie d imension . T h e n there ezist a finitely generated sub;nodule A

of A4 and a submod,ule B of A such that M / B is finitely embedded and .4/B i s s emi s imp le ar t in ian .

Proof. \Ve %,ill construct inductively t\vo ascending sequences of submodules

of 11, {.-1,),20 and {B,),lo such that for any 1 > 0 we lvill h a w

(1) A; is f.g., B, A, andfor i > 0; B, C A,.

(2) .?,/B, = Soc(M/Bi).

(3) n B ~ + ~ = B,

(4) At+l/'B,+l Ai/B, 8 Ai+l/(At + B,+I) . ( 5 ) A,+B,+l = Ai+1 if and only if .A, = .A,+1 if and only if Soc(.'M/B,) <,

M I B , .

Set A. = Soc(k1) and Bo = 0, and notice that since iLf has finite Goldie dimension A. is finitely generated. Assume we have '40 . . . :; Ai and

Bo C . . . C B, satisfying conditions (1) to (5). If Soc(M/B,) 5 , h l / B , we

take A;+1 = A, and Bi+1 = B, . Otherwise there exists m E ilf \ A, such

that

Let N be a maximal submodule of mR containing mR f l B i , set B,+l =

N + Bi and Ai+1 = T-'(Soc (M/Bi+l) ) where T: 1W -+ M/B,,1 denotes the natural projection.

It is clear that Bi C Bi+l, Ai+l/B,+l = Soc(M/B,+i) and B,;, C & + I .

It follows from the construction that if Soc(M/Bi) 5, M / B , then Ai = A,+1, to prove the converse, observe that if Soc(.Vl/B,) is not essential in M / B , then m E A;+, \ A;.

We also have that if A; + B;+l = .4i+l then M/B; has essential socle, for otherwise

but this is impossible because of the election of m and LV. This proves that condition ( 5 ) is satisfied.

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1220 FAITH 4ND H E R B E R A

If .-1,/B, 5 , XflB, then it is easy to check conditions (I j to (4). Froni now on we will assume tliac Mi B, does not have essential s o c k

By the construction we have that B,+l n '4, C (mR + B,) n A , = B,, since

B, C .A, we have tha t B,+1 n A, = B, hence condition ( 3 ) 1s satisfied

thus ..'ci,y;il is semisimple and we have A, C Since S o c ( ~ \ l B ~ + ~ ) is semisirnple the exact sequence

splits and h g A @ " ' + I

BStl B, A , + B ; + l l so (4) is satisfied.

Since M / B , + l has finite Goldie dimension, S O C ( M / B , + ~ ) is finitely gen-

erated and A,+] = mlR + . . . + m,R + Bi+l for suitable m l . . . . , rn, E :\.I.

By definition Bi+l = N + B , C mR + Bi C mR + Ai, since

mR+ A, N

mR ‘4i mR -4, - $-=-F-

Bi+l m R n B , + , Bi N " B ,

is semisimple, we have mR + A, A,+l and we may conciude that A,+l =

m l R + . . . + m,R + mR + A, is finitely generated. So (1) is satisfied and this

finishes the construction of the two sequences.

Set B = U,,o B , . For each z 2 1 we have -

By (3) A,+l n Bi+l+l; = B,+I for any k 2 1, thus "'i;l+B 2 %+i B,+I By (4)

Since M/B has finite Goldie dimension there exists io such that

for any k > 1. By ( 5 ) this implies that .4,, = A,o+k for any k 2 0, and by

( 3 ) also B,, = B , , + k . Hence B = B,, and by (3) M I B has essential socle. Taking A = A,, we have by (1) that A is finitely generated and by (2 ) t h a t

.4/B = S o c ( M / B ) is semisimple artinian. O

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LINEARLY COMPACT MODULES 1221

Corol lary 1.2. (Enochs (En], Zosch~nger [Zoj) Let R be n commutatzve noe-

therzan rzng iL1 za a n R -modu le such t ha t a n y quotzent oJ .\I haa jinzte Goldze

dzmenston zf and only zf ?il zs a n ez tenszon of a finztely generated 'by a n ar-

tznzan module 0

Let ;if be a right R-module. .A finite family ill,. . . , A n of proper submod-

ules of .Vl is said to be coindependent if for each i, 1 5 i 5 n; .A, f ['l,+, A, =

2V1: and a family of proper submodules of M is said to be comdependent if

each of its finite subfamilies is coindependent.

It can he shorvn that there exists a maximal coindependent iamily of

proper submodules of :bI. If this family is finite then its cardinality is uniquely

determined and is called the dual Goldie d imens ion of .\I, then A4 has finite

dual Goldie d imens ion . If the family is infinite then 1\4 has infinite dual

Goldie d imens ion .

A module with dual Goldie dimension 1 is said to be hollow or couni form

and a cyclic hollow module is said to be local. We refer to [Val and [Ha-S]

for the general facts about the dual Goldie dimension.

It was proved by Zelinsky [Z, Proposition 61 that a linearly comr~act mod-

ule has finite dual Goldie dimension. Miiller MU^]) showed that if ]\/I is a

linearly compact module then there exists a finite family of hollow submod-

ules of M, (HI:. . . , H,) say, such that iV1 = HI + . . . + H,. For the kind of problems we will be looking at we can always as:.ume that

the ring is local as it is shown in the following lemma mostly cosltained in

[H-S, Lemma 81.

1.3 Lemma. Let R be a commuta t i ve r ing and M a LC. module t h e n EndR(M) is semiperfect and there exists a finite fami ly of diflerent m a z i -

ma1 ideals of R, :MI,. . . , M , say, such that :

(1) M = iVIl $ . . . @ Id,, where each .M, zs 1.c. as a n R M , -module and as

R -modu le .

(2) End.q(M) 2 EndRMl (MI) $ . . . $ end^," (M, ) . (3) 1l.l & R M 2 (hI1 MI) @ . . . $ ( M n € 3 ~ ~ ~ ~ M n ) .

Proof. B y [H-S, Corollary ?I] EndR(;b1) is semilocal (even for a ncncommu-

tative R ) , since linearly compact modules over a commutative ring are pure

injective (c.f [J-L, p.174 and Corollary 8.271) we deduce that ErLdR(.\I) is

semiperfect.

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1222 FATTI1 A N D HERBERA

From [H-S, Lemma 8] we know that M = $,EIM, where 41, are R,M,- modules for certain maximal ideals jut of R. Since :lI is 1.c. it has finite

Goldie dimension, thus I is finite and being 11.1 1.c. as R-module each ibl, is

1.c. as R , M , - m o d ~ l e . Also from [H-S. Lemma S] x e know that ( 2 ) holds.

LVe claim that iW, @R AdJ = 0 for z # j. From the proof of [H-S: Lemma

81 we know that for any x E !%I,, R/r (x ) is a local ring with maximal ideal

;M, / r ( s ) . Herice , M I is the unique masinial ideal of R containing r ( s ) . Iii particular there exists r E r ( z ) \ ;!AJ. Since :\IJ is an R,w, -module. for any

y E :\)IJ y = y l r for some y1 E MJ thus a @ y = z 82 y'r = 0 for any a E JU,.

y E .\I, and the claim follo~vs.

By using the uniqueness of the tensor product of two inodules it is easily

seen that for any 2 . !\dl 8~ .bI1 'I, -21, @ R ~ , M, which establishes (3) .

A right linearly compact ring R (commutative or not) is semiperfect

(cf. [Os]), as x e just saw this is still true for the endomorphism, ring of a

linearly compact module over a commutative ring. In general the endomor-

phism ring of a linearly compact module is semilocal (cf. [H-S]) but not

necessarily semiperfect (cf. [C-!dl, [C-F] or [F-H] for counterexamples). This

implies that the endomorphism ring of a linearly compact module is not, in

general, linearly compact.

1 . 2 L inea r topologies: t h e !\I-topology. In this subsection we will see

that to compute the endomorphism ring of a module we can assume that

the ring R is complete in the M-topology. We will make strong use of this

property in $ 5 and $6.

Let R be a ring (commutative or not) and Ad a right R module. Let

{Mi)iEI be an inverse system of submodules of M, we say that M is com-

plete in the linear topology induced by if the natural map M --+ @M/iVI, is onto. Equivalently M/ nGI Id, E @Ad/.\&. Notice that h.1 is complete in the topology induced by {Mi)iEI if and only if any finitely

solvable system of congruences

has a solution in M . Equivalently, if n , E F ( x , + M,) # 0 holds for every finite

subset F C I, then it holds for F = I The topology induced by the {- \ / I , ) IE~ is Hausdorff if and only ~f n ,EIMl =

0 The H n u s d o r f f completzon of .L1 In the llnear topolog~ !:iduced by {-lIl)tEr 1s I@M/A!IL

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LINE:\RLI' COMPACT MODlrLES 1713

Kow let R be a com~riurativc ring and .\I a r.i;i;t R- nodule. Tile +\I-

topology "11 R is the linear topolog~. having the set of all the ke~.n,:ls of ho-

nionlorpiiisrns froin R into .'PI< k 2 0, as basis of open neighborhoods of

0. Denote bg. R the Hausdorff completion of R in the .\I-topolo~,y. Then

R/ annn(d l ) is a dense subring of R, in fact if hl is finitely gener:.ted then

Rj amiKj.I1j is the completion of R in the .\i-topology. .\I has a :anonical

structure as a module over R. since for any m E ILI and I' E R there exists

r E R n (I' + annR(m)) and we define ? m = r m .

Let .\I be an R-module we denote by C R ( N ) the set of all R-sul~modules

of :\I. If R is the completion of R in the .'PI-topology then LR(* l l ) = Lii(-\.I).

1.4 L e m m a . Let !\I and iY be modules over a commutative rzng !2

( 1 ) Let R be the completzon of R zn the .\4-topology then Enci~(.\.I) =

EndR(.\l) and Lj7(M) = CR(.\.l) ( 2 ) Let R be the completzon of R zn the M@!V-topology T h e n -11 ER.V 2

-11 sR iY as abelzan groups and L R ( M @ R .V) = LR(>\/I 9, AV)

Proof. Statement ( 1 ) follows from the remarks preceding the lemma.

It is easily seen that if G is an abelian group and f : ~\f x -V i G is

Z-bilinear, then f is R-balanced if and only if it is R-balanced. This proves

that i1.I 2, N and M @,Q 1V are isomorphic as abelian groups. k For any 3: E M @R N , 3R = ZR: since if a: = xi=, rn, @ n, and ? E R

then there esists r E R n (+ + annR(rnl, . . . , m k , nl.. . . . ni,)) and fx = r x .

Thus C R ( M @ R N) = L R ( M @fi N ) .

1.3 Linear ly c o m p a c t modules and M o r i t a duality. One of the con-

texts where linearly compact modules have an important role is in the Morita

duality theory. We quickly summarize some general facts about lIorita du-

ality and we refer to [XI for a complete development of the theory

Let R be a ring (not necessarily commutative), E an injective ccgenerator

of the category of right R-modules and M a right R-module. The E-dua l module of M is M* = H o r n ~ ( M . E), iM* is a left module over the ring S =

EndR(E) . Consider M** = Homs(M, E) , since E is an injective cogenerator

M is naturally embedded in Ad**, we say that the module is E-refiexive if

the natural embedding is an isomorphism.

.\ ring R (commutative or not) has a right Morita duality !l\lo] if R has a

duality functor HornR( , E ) induced by its minimal right injective cogenerator

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1223 FAITH A N D HEKBERA

E. hluller in [Llu?] proved that R has a right Llorita duality if' and only if '

Rn and ER are l .c, modules, moreover in this s i t ~ ~ a t i o n the category of 1.c.

modules coincide with the category of E-reflexive niodules.

3Iiiller also showed [Mul l that if R is commutative then R has a hlorita

duality if and only if E induces a Morita self-duality. i.e., R = E n d R ( E ) .

This result kvas impro\.ed by Anh, giving a positive answer to a long standing

conjecture:

1.5 T h e o r e m . ( h n h [.A]) Let R be a commutat ive 1.c. ring then its minimal

injective cogenerator E i s linearly compact. Th,us if R is a commutative 1.c

ring zt has a Morita self-duality.

1.6 L e m m a . Let R S be a n extension of rings and let E be a right injec-

tive cogenerator of R then S* = HornR(,!?, E) is a right injective cogenerator

of s. 1.7 Proposition. (Vbmos [V2, Theorem 2.21, [XI Theorem 7.61) Let R be

a commutat ive 1.c. ring and let E be its min imal injective cogenerator. If S is a linearly compact R-module, then S has a self-duality induced by the

S-module S * . Moreover if a module Ms is 1, c. then MR is 1.c.

Theorem 1.5 and Proposition 1.7 are two key results for the rest of the paper , we will make constant use of them a n d many times without previous

acknowledgement. By Theorem 1.5 a commutative linearly compact ring always has a bIorita

self-duality and by Proposition 1.7 whenever we prove that the endomor-

phism ring S of a linearly compact module over a commutative ring R is

R-linearly compact, we have tha t S has a Mori ta self-duality (c.f Corollary

1.9).

1.8 Lemma. Let R be a 1.c. ring and N a 1.c. module. T h e n

Proof. We have the natural isomorphisms

where E denotes the minimal injective cogenerator of R. 0

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LlKEARLY COMPACT MODULES 122.5

1.9 Corol lary . If :\,I zs a i l R,-7nod.ule over (1 1.c. ring R, such that A =

E c d E .\I is R-1.c.; then A has a self-duality i n d ~ ~ c e d by -4' = Hor:lR(A, E) where E denotes the min imal injective cogenerator of R. Moreover

zs the minzmal znjectzve cogenerutor over A, and A 2 EndA A* .

Proof B y Lemma 1 8, A S (-1.1 &?R ~v*)', and A* Z M E R LbIl/l' From Propo-

sition 1 7 we bnon that A* induces a hIorita self-duallty hence EndA i .4* ) 2 4 and '-1' is the mlnlmal in jec t l~e cogenerator of '4 0

1.10 T h e o r e m . Let ,Id and 1V be R-modules If IM zs a jnztely generated

R-module , and N zs 1 c , t hen HOmR(!bf. ,V) and .'d E R 4' are both 1 c

Proof. The exact sequence

is converted by H o m ~ ( , IV) into an exact sequence

so HornR(!Lll. .V) is 1.c. since N n is 1.c.

Dually, the epimorphism Rn + M gives an epimorphism

so M @ R iV is 1.c. since iVn is. 0

1.11 Corol lary . Let R be a ring and h/l af ini te ly generated Eznearly compact

module. T h e n R/ a n n ~ ( M ) is a linearly compact ring.

Proof. Since M is finitely generated R / a n n R ( M ) L, M n and therefore

R/ annR(l%f) is linearly compact.

1.12 Pa r t i a l D u a l T h e o r e m . If R is LC., and !V is finitely embedded, then

HomR(&f, N) is 1.c. for any 1.c. R-module M.

Proof. Since R is LC. by Theorem 1.5 its minimal injective cogenerator E is also 1.c. as R-module. Since N 2 En, iV is also 1.c. as R-module. By Lemma 1.8

(hf @ R !Vi)* 2 HomR(kf! :v),

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1226 FAITH A N D HERBERA

and since .V* is finitely generated by Theorem 1.10 (~bl 8 1 3 :I-* j is l.c., hence

(:If zii :\:*)* is also 1.c.. as needed. C?

When ibf is a finitely generated linearly compact R-module, we saw in

Coro l l a r 1.11 that R = R/ a n n ~ ( M ) is a linearly compact ring. One sees

that E = anns (annn(M)) is the least injective cogenerator for mod-R, hence

induces a self duality for R . Obviously, A = EndR 31 = EndR ,MI and also

\.Ye show now that rational estensions of finitely generated modules have

also 1.c. endomorphism ring.

1.13 T h e o r e m . If 121 i s a n R-mod,ule and if H ~ r n ( ~ \ d / N . M ) = 0 for a

j n i t e l y generated submodule !V, then A = EndR M embeds an HomR(lV, M) canonically qua R - m o d u l e . T h u s zf IVJ is R - L C . , t hen so is A.

Proof. HornR( ,Ad) converts the exact sequence

into the exact sequence

of R-modules, and HomR(l\d/lV, 1VI) = 0 implies the desired embedding. The

theorem then follows from Theorem 1.10.

1.4 Rings satisfying ERC.

1.14 IRC Theorem. T h e following are equivalent condit ions o n a c o m m u -

ta t ive LC. r ing R.

( 1 ) E v e r y 1.c. module M has R-1.c. endomorphism ring.

( 2 ) HomR(M, N) i s 1.c. for all 1.c. modules M and N.

(3) M @R IV i s 1.c. for all 1.c. modules M and N. ( 4 ) If M and N are finitely embedded linearly compact modules, t h e n

M @ R iV i s l inearly compact. (5) If M and N are linearly compact, :Vl is finitely embedded and N

is f ini tely generated. T h e n H o m ~ ( h f , IV) is a lznearly compact R- module .

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Prooj Let and .Y he R-modules Since

as R-modules, it follows that (1) and (2 ) are equivalent. Since R is linearly compact, the dual of a linearly compact modv.le is also

R-linearly compact, thus Lemma 1.8 shows that (2) and (3) are equivalent statements.

(3)+(4) is clear and (4)+(5) follows from Lemma 1.8 and frorr: the fact

that the dual of a finitely embedded module is finitely generated.

\Ye have to show that (5)+-(1). .lssume that lV1 is a nonzero linearly

compact R-module, by Proposition 1.1 there exist B C_ -4 ll/I such chat A is finitely generated, A / B is semisimple artinian and :\4/B is finitely ernbedded.

If we apply the functor Hom(k1, ) to the sequence

we get

By Theorem 1.12 we know that Horn(-VI, M / B ) is 1.c. thus Horn(!\{. ]\I) is 1.c. if and only if Hom(M, B) is 1.c. Since B C A we have that HomjM, B) is LC. provided Hom(M, A) is.

If we apply the functor Hom( , A) to

we get 0 -+ Hom(M/A, .4) -+ Hom(M, A) -+ Hom(A, A).

By Theorem 1.10 Hom(A, A) is 1.c. thus Hom(M, A) is linearly compact if and only if Hom(iM/A, A) is.

If we apply the functor Hom( , A ) to

0 -+ A/B --+ MIB i M I A -+ 0

we get

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1228 FAITH A N D HERBERA

By Theorem 1.10 Hom(.4/B. A ) is 1.c.. thus Hom(.U/.4, '4) is linearly com-

pact if and only if also is Ho~n(:lI I B: .4). Since A is finitely generated and J l / B is finitel!. embedded, by hypothesis

Hom(:\6/B, A) is i.c., tlius Horri(A1, .\.I) is linearl!. compact for any linearly

compact R-module M. 0

If R is a commutative 1.c. ring satisfying the equivaleiit statements of

Theorem 1.14 Lve say that R sntisfies ERC.

Linearly compact modules over an artinian ring are finitely generated

(cf. [S, Corollary 3.51). so Theorem 1.10 sho~vs that x t in ian rings satisfy

ERC. It is easy to see that complete noetherian rings satisfy ERC verifying

statement (5) in Theorem 1.14, but we will wait until $6 to see that and to be

able to draw more properties on the structure of linearly compact modules

over noetherian rings. We will show in 53 that another class of linearly compact rings satisfying

I R C are masimal valuation rings.

Proposition 1.7 yields that any 1.c. module over a 1.c. estension of a ring

satisfying &RC has l.c, endomorphism ring:

Proposition 1.15. Let R be a 1.c. ring satisfying ERC, S an R-1.c. R-

algebra (not necessarzly commutat ive) and A4 a right linearly compact S - module. T h e n Ends(,V) is R-1.c. O

In particular we will have that finitely generated algebras over local com-

plete noetherian rings or maximal valuation rings satisfy ERC. Also a trivial

extension of a ring satisfying ERC by a linearly compact module will satisfy

ERC . The R-dual of :VI is denoted M' = H o ~ R ( - ~ , R). We note

1.16 R-Dual Theorem. The R dual of M over a 1.c. ring R with minimal

injective cogenerator E , i s

h I 1 2 (1VI @'q E ) *

hence is R-l.c. i$ &1 lpR E zs LC. Furthermore ERC implies that M' is 1.c. /OT

all 1.c. M.

Proof. Since R* = E , by Lemma 1.8 lkf' = Hom~(!kf.R) 2 (?vf @ R E)* hence M' is 1.c. iff (A1 3~ E)* is 1.c. iff M @R E is 1.c. Since ERC implies

by Theorem 1.14 that M @R !V is 1.c. for all 1.c. R-modules PI, AV, the last

statement holds 0

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,4 ring is reduced if 0 is the only nilpotent element, i.e., if it is a semiprime

ring.

2.1 L e m m a . If R is a reduced 1.c. r ing, t h e n i t s quotient ring Q is a finite

product of fields and coincides wi th the m a z i m a l ring of quotients af R.

Proof. The maximal quotient ring Q* = Q,,,(R) is ])'on Neumanx regular,

since R has finite Goldie dimension. Q2 is a finite product of fidds? and,

moreover Q = Q2. (cf., e.g. [FS, proof of' Theorem 3.31).

A module i'l/l is tors ion free (= t.f.) if no element m + 0 in X I annihilates

a regular element of R. Since by the proof of Theorem 2.1, any 1 . ~ : . reduced ring R is semiprime Goldie and every essential ideal of R contains a regular element. It therefore follows that the singular submodule of a motlule IV

Sing(N) = {m E IVI annR m < R} ess

coincides with the torsion submodule t ( N ) consisting of all m E A T annihi- lated by some regular element of R. Moreover, N / t (N) is t.f.

2.2 Examples .

1. Any torsionless module N over a reduced Goldie ring R is t.f.

2 . Any R-dual !V1 = HomR(N, R) is torsionless, hence t.f

3. Over any reduced ring R, any indecomposable nonsinguhr module

N embeds in Qz = Q,,,(R) (Goodearl [GI, p. 19, cf. [F'2]. p. 86,

Ex. (q)) , hence over a reduced 1.c. ring R, any t.f. indecomposable module embeds in Q (since Q = Q2-see Lemma 2.1).

2.3 Theorem. Le t R be a reduced 1.c. ring and M a n R-LC. t c m i o n free

module. T h e n A = EndR(M) and n/l' = HornR(M, R) are R-1.c. modules.

Proof. Let E ( M ) denote the injective hull of iM. Since M has finite Goldie

dimension,

E ( M ) = El . . . $ E n

and for i = 1 , . . . , n, Ei = E(Ui) is the injective hull of a uniform cyclic submodule Ui = m,R # 0 of 1l-I. By Goodearl's theorem mentioned in

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1230 FAITH A N D HERBERA

Examples 2.2.3! E, embeds in Q for any i. Hence E(.\I). kvhence .\/I embeds

in Q" for some 71. < CO.

Since by Lemma 2.1 Q n is injective. Q n E E(-\.l) 5 T. T n R n # 0 , and

&I /I (T n R n ) is and essential R-submodule of Q" because Q is the ring

of quotients of R. ,%I % (T n R n ) is R-1.c. because .If and R are. Since EndR(h/f) and Hom~(!bf, R) are R-submodules of E n d ~ ( i \ f $ (T n R n ) ) and

HomR(nf $ (T n R n ) . R ) respectively: it is enough to prove the statement for

an essential 1.c. submodule of Qn . Since by Lemma 3.1 R is a semiprime Goldie ring, M contains an essential

submodule F S R n . Hence we have the exact sequence

Since M and R are t.f. HomR(M/F. 121) = 0 and HomR(M/F. R) = 0, hence

and

0 -+ H o r n ~ ( M , R ) 4 Hom(F, R)

By Theorem 1.10 E n d ~ ( h f ) and Hom~(!\f. R) are R-1.c. 0

In this section we prove that maximal valuation rings satisfy E R C (The-

orem 3.3) and we generalize Vbmos' theorem [V2] that states that any inte- grally closed domain with R-1.c. quotient field Q ( R ) is a valuation domain

(hence a MVD) to reduced rings with linearly compact quotient field, see

Theorem 3.6.

3.1 Proposition. Let M and 121 be uniserial modules over a ring R.

( 1 ) If R is L C . then Horn~(iZ/I, N ) is uniserial as R-module .

(2) (IS-L, Theorem 2.3 and Theorem 3.31 or [F-S, §VII.2]) EndR(.bI) = R~ where R denotes the completion of R i n the M-topology and P is

a certain prime ideal of R. (3) i\/l @ R N is uniserial.

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LINEARLY COMPACT MODULES 123 1

Proof. To prove (I), let f and g be elements of H o r n ~ ( M , iV): and consider

;\/II = {m E M(,f(m)R C g(m)R) and &I2 = { m E A1igjm)R C J[r.a)R). lbll

and iV12 are submodules of M because hf is uniserial. Since N is uniserial u 1\12 = :Vl and since M is uniserial either ,MI = M or id2 = ibf.

Assume that MI = Ad. then for any m E M there exists r , E R such that f(m) = r,g(m). Consider the system r E r , ( a n n ~ ( g ( m ) ) ) and notice that it is finitely solvable, whence solvable. If r is a solution then f = rg and so

f R gR. If &I2 = &I we can conclude that gR 5 J R , thus HomI,:(M, N ) is a uniseriai module provided that R is 1.c.

To prove (3) observe that if ml 9 n l and m? 8 nz are elements in :\I 3~ 1V Lve may assume that ml R 2 m 2 R thus r n l 8 nl = mar @ nl = ma ;? 71; . NOW either n2R C n;R so m2 @ nzR mi 8 n lR , or n;R 2 n z R which yields m l @ nl R C ma @ n;? R. Hence iV @ R IV is uniserial.

3.2 Lemma. A n y uniserial module over a 1.c. ring is 1.c.

Proof. Let M be a uniserial module over a 1.c. ring R, fix 0 # x E M and

let iV be a maximal submodule of xR. Then M/!V is uniserial and finitely embedded. Since R is 1.c. M I N and !V C_ xR are L C . , thus M is 1 c. 0

3.3 T h e o r e m . ERC holds for maximal valuation rings.

Proof. Let R be a maximal valuation ring and M a linearly compa.ct module,

we want to show that EndR(M) is also linearly compact. Since :\I has finite

Goldie dimension A1 v El @ . . . $ En, when each E, is indecomposable in- jective. By [F2, Theorem 20.491 each E, is uniserial, thus 1.c. by Lemma 3.2.

Since HomR(M,h/l) 2 $:=, HomR(kf ,E, ) , it suffices to show that for each i, HomR(M, Ei) is linearly compact.

Since Ei is injective

HomR(Ej ,Ei) is uniserial by Proposition 3.1 hence linearly compact by Lemma 3.2, thus HomR(lM, Ei) is linearly compact.

3.4 Corollary. ([Gi, Lemma 2)) If R is a MVR, and P is a prime ideal, t h e n the local ring R p is LC. as R-module and moreover

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FAITH AND HERBERA

In particular, Q(R) is 1.c

Proof. R p is a uniserial module over a 1.c. ring thus by Lemma 3.2 it is

linearly compact as R-module. Set E = E ( R / P ) . Then E is canonically an Rpmodule ,

and R p E EndR, E = EndR E because R p is a h1VR. This completes the proof of the corollary since the set Po of zero divisors

of R is a prime ideal (in any uniform ring) and Q ( R ) = Rp,. 3

3.5. V6mos ' F i rs t T h e o r e m . ([W, Proposition 2.11)) If R is a domain

with 1.c. quotient field Q(R), then the integral closure S of R i n Q(R) is a

valuation ring.

3.6 T h e o r e m . If R is a reduced ring integrally closed i n Q(R), and if Q(R)

is R - L C . , then R is a finite product R1 x . . . x R t of MVD's , and Q(R) is a

finite product Q(R1) x . . . x Q(Rt) o f fields.

Proof. Since Q(R) is 1.c. as R-module so is R. By Lemma 2.1 Q(R) =

Q1 x . . . x Q,? where Qi are fields. Since R is integrally closed in Q(R), then R contains each projection idempotent e, : Q(R) -i Q,, i = 1,. . . , n: hence R = R1 x . . . x R,, where R, = eiR is integrally closed in the field

. . e,Q(R) = Q, , z = z , . . . , n . By V&mosl First Theorem 3 .5 Ri is a valuation domain, hence an MVD and this completes the proof.

The next result is a slight generalization of Vamos' First Theorem 3 . 5 (= the case where Q(R) is R-LC.)

3.7 T h e o r e m . If R is a domain with quotient field Q, and if at least one

overring A of R contained i n Q is R-1.c. and a VR, then the integral closure

S of R i n Q is a VR, hence a maz imal valuation ring.

Proof. Vamos [V2] proves this assuming Q R is 1.c. but a look at the proof

shows that our hypothesis suffices: what is needed is that the valuation overrings {V,) of R are comparable. That is, that either V, > V, or V, > V, for all i # j . The proof appeals to Theorem 107 in [K] which states that non-comparable valuation overrings Vl and V2 are localizations at different maximal ideals Mi and 1% of Ro = Vl n V2. If we take Vl = A, then we see that Ro = A n Vz is 1.c. since A is 1.c. hence & is a local ring. and therefore

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LINEARLY C O M P A C T M O D U L E S 1233

.A and 1.5 are comparable. It follows then that S is the intersection of the

chair1 {V, j I E J . But this impiies that S is itself one of the I,;, since, 5.g. either q or q - i E S for all q E Q . O

I'8n1os gi\res a complete characterization of complete noetheria? domains

with 1.c. quotient field:

3.8. VBmos' S e c o n d T h e o r e m . ([V2, Theorem 3.71) If R is a Noetherian

d o m a l i ~ , t h e n Q ( R ) i s 1.c. iff R zs a field, or a complete local ,domazn of

d imens ion 1 .

'CC'eakley in [W] studies almost finitely generated (=a.f.g.) modul'es, that is

modules that are not finitely generated but any proper submodule is. VQmos

and M'eakley's results can be seen together in the following proposition:

3.9 Propos i t i on . Let R be domain t ha t is no t a field and denote b y Q i t s

j e ld of quot ients . T h e n t he following s ta t emen t s are equivalent:

(1) R is noe ther ian and Q is 1.c. as R-module .

( 2 ) QIR is a r t i n ian as R-module and R is a complete noetherian rzng.

( 3 ) R is a complete noether ian d o m a i n with Krul l d imension 1 .

( 4 ) R is a comple te noether ian d o m a i n wi th Krul l d imension 1 and i t s integral closure i s a discrete valuation ring that is finztely generated

as R - m o d u l e . ( 5 ) Q ( R ) / R i s a n a r t i n ian a.f.g. R -modu le and R is a 1.c. r m g .

(6) Q(R) is a n a.f.g. linearly compact R-module .

ProoJ. Statements ( I ) , ( 2 ) and (3) are equivalent because of VQnlos' result

and its proof, [V2, Theorem 3.71. Statements (4),(5) and (6) are equivalent because of Weakley's [W, Proposition 1.41 and V6mos' results.

The t ru th of the proposition follows from the fact the if R is a one- dimensional complete noetherian domain, then Q ( R ) / R is an artinian a.f.g. module (cf. [M3, Theorem 7.11). 0

Let Ii' be any field and for i = 1 , . . . , n, let cu, E N. The power series ring

K [ [ x Q : , . . . , x O n ] ] is a complete noetherian domain with Krull dimension 1

whose integral closure is finitely generated as R-module.

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1234 FA17'I~I AND H E R R E R A

In this section we study an indcconiposable 1.c. R-rnodulc .\I and its erldo-

morphism ring A ( a local ring with maximal ideal iM) , and show, i.a., that

when ,1.I is faithful then R I P has R-1.c. quotient field where P = ,U T! R is a prime ideal. ERC implies that in addition the local ring Rp of R is

necessarily R-1.c.

4.1 P ropos i t i on . Let .I1 be a nonzero zndecomposable 1 c. module over a

rzng R and let R = R/(annR .LI), t hen

(1) EndR(l\/l) zs a local ring.

( 2 ) RP EndR(-U) , .where P = ;M n R and ;M denotes the m a z i m a l

ideal of A. (3) M is a n R p - m o d u l e and a n R p - m o d u l e where P is the pullback pr ime

of P. (4) Q ( R / P ) = &(RIP) = R ~ / P R ~ is 1.c. over R. I n particular the

integral closure of R I P i n Q ( R / P ) is a maz ima l valuat ion r ing .

(5) T h e se t of zero divisors oJ R is contained i n P . ( 6 ) If EndR(iZ/I) i s 1.c. over R, t hen R p is 1.c. over R.

Proof. ( 1 ) holds because by Lemma 1.3; EndR(hf) is semiperfect, hence local

by the indecomposability of M. Obviously R L+ EndR(h.I), and s-' E EndR( lu ) for all s E S = R\P,

hence Rp = R S - ' C E n d ~ ( h / l ) . (2) ,(3) ,(5) and (6 ) follow immediately.

E R ( R p / P R p ) is the minimal injective cogenerator of R p , thus

and the simple submodule of E ~ ( R p 1 p R p ) is contained in an homomorphic

image of M, hence it is linearly compact as R-module. Since this simple

subrnodule is isomorphic to Q ( R / P ) = & ( R I P ) = R ~ / P R ~ statement (4)

follows. To prove that the integral closure of R I P in & ( R I P ) is a maximal

valuation ring just apply Theorem 3.5. 0

P can be a maximal ideal, this is the case if $1 has nonzero socle or a

maximal submodule, cf. Lemma 4.10.

The next corollary follows from the last proposition. and Proposition 3.9.

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LINEARLY COMPACT MODULES 1235

4.2 Corollary. If ( i n the above) R is a Noether ian rzng, t hen e2th.r R I P zs a field or a complete Noetherzan local d o m a i n of dzmenszon 1 . h'ence zn

this s i tuat ion the integral closure S of R I P in & ( R I P ) is a complete discrete

local ring finitely generated ovcr R I P .

4.3 Corollary. Il, w i t h ~ r ~ the i~otatzorz of rropoditiorz 4 . i . .-I = Erld,<(.lIj is

1.c. over R, t h e n i ts r e s i d ~ ~ e s keu f i e ld I< = .-1//M has finite dimensior, . [I< : k ]

over the reszdue field k = Q(R/P) of R p . Moreover, if R p has algeiliraically closed residue field, t hen I\' = k and A = R p + J M . Flnnl ly , if Rp i s a n

algebra over k, t hen

.-I =k+,M

Proof. Since A is 1.c. over R. then A is 1.c. over the local ring B = ,qp , and

consequently I< = A/;M is 1.c. over k = B / P B , since P B = J M ~ ~ B . Ciince 1.c.

modules have finite Goldie dimension, the vector space dimension [.K : k] < m. Thus, if k is algebraicaliy closed, then K = k j whence A/JMA = B I P B , so A = B + J M . The rest follows forthwith. 0

4.4 Proposition. Let R be n 1.c. ring and P a pr ime ideal. T h e n E ( R / P )

is LC. as R-modu le if and only if R p is R-1.c.

Proof. Assume that E ( R / P ) is 1.c. as R-module. E ( R / P ) is the minimal

injective cogenerator of R p , and the R p module El = Hom,q(Rp, E ) (where

E denotes the minimal injective cogenerator of R) is also an injective cogen-

erator of Rp . Thus

E ( R I P ) 9 El (*) .

We have that,

Thus we may conclude that Rp E ( R / P ) * as R-modules. Since R and

E ( R / P ) are R-linearly con~pact, E ( R / P ) * is R-linearly compact and we may

conclude that R p is R-linearly compact.

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1236 FAITH A N D I i E R B E K A

If R p is linearly compact over R then it is linearly compact as a ring. thus

by Theulwri 1.3 E j R j P ) is Rp-linearly compact and by Proposition 1.7 also

R-linearly compact. O

Let .b1 be a module over a (possibly noncommutative) ring R and let S be

a 11ng such that its elements act on ILL as endomorphisms, we may consider

the subset of S of zero divisor of ,\/l

Z s ( M ) = { f E S/ f is not injective)

aiid d ~ d l l y

Lli,(hI) = { f E S/ f is not onto}

If R is commutative and EndR(h.l) is local with maximal ideal .M, then the

pull back prime P of R/ a n n ~ ( M ) n M is P = Z R ( M ) U WR(M) . Some more

information about P can be obtained when the module 11.1 has finite Goldie

and dual Goldie dimension.

4.5 P ropos i t i on . Let R be a (possibly noncommuta t iue ) ring, 11.I a nonzero

rzght R -modu le and S a subrzng o f E n d ~ ( A 4 ) .

(1) If ;\.I i s a ,uni form R-modu le t h e n Zs(.VI) is a completely prime ideal

o f S . ( 2 ) I j :lI is h o l b w t h e n W s ( M ) is a completely pr ime ideal of S .

Proof First we prove (1). Assume that f and g are nonzero elements in

Zs(M). since iL1 is uniform Kerf n Kerg # 0 thus f + g E Zs( !W) . If 0 f j E Zs(M) then it is clear that for any 0 # g E S , g f E Zs(!Lfj; since

M is uniform Kerf n Img # 0 thus we also have that f g E Zs(M). Assume

that f g E Zs(M), then it is clear that either f E Zs(M) or g E Zs(.bf).

Thus as we wanted to see, Z S ( M ) is a completely prime ideal of S. Dually for ( 2 ) . 0

4.6 Corollary. Let R be a (possibly noncommuta t zve ) rzng and kl a n R- module with local e n d o m o r p h i s m ring and w i th Goldie d imenszon 1 and dual

Goldie d imens ion I . T h e n e i ther a n y injective endomorph i sm is onto or any

on to endomorph i sm is injective.

Proof. Apply Proposition 4.5 taking 5' = E n d ~ ( l W ) , then the maximal ideal

of S is Zs(41) U T.Vs(h.I). But Zs (h f ) U W s ( M ) is an ideal if and only if

Zs (M) C Ws(M) or Ws(M) C Zs(11.l) as we wanted to see.

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LINEARLY C O M P A C T MODULES 1237

c :ruserial ' - modules are examples of modules with Goldie and dual Goldie

din~ension 1. .Any 1.c. nloclu!c can be ernbedcied in a sum (possib!~ no:: direct)

of modules ivith Goldie and dual Goldie dimension 1.

From now on rings ivill be again commutative

4.7 L e m m a . Let R be (I rLng and 11 , *V be R-modules

( 1 ) I f f -+ .V zs an zn~ectzue morphzsm then ZR(.Y) > Z R l r l l ) , z f zn

addztron the zmage o,f J zs essentzal zn .bI then Z R ( h l j = Zf, (.\-) ( 2 ) If f ,\I 4 -1- zs an onto morphzam then bVR(.V) C ILYRl J I j ~f zn

addztzon the kernel of zs small zn then W R ( M ) = WrR( V)

Proof, The proof of (1) is routine. If f : h --+ 117 is onto, it is ea:..); to see

that W R ( , V ) C_ WR(lZ/I). Let r E W R ( M ) l if r N = IV then hf = r.M + K e r f

and since Kerf is small in iM we have that r.Vl = ,bI which is impossible.

thus r E W R ( A T ) 0

If a module i\/l has finite Goldie dimension n, then it contains an essen-

tial finite direct sum of uniform submodules, that is U1 $ . . . $ C',, 5 , M , thus by Lemma 4.7 Z R ( M ) = Z R ( U 1 ) U . . . U Z R ( U n ) . B y Propo~ition 4.5

{ Z R ( U i ) ) , = l , , . , , n is a family of prime ideals of R, and we have:

4.8 Lemma. Let iVf be an R-module ,with finite Goldie dimenszoq n such that U1 @ . . . $ U , <, M, and each Ui is a uniform module. Then there

exists { i l , . . . , i k ) 2 { I , . . . n ) such that Z R ( M ) = Z R ( U ~ ) U . . . U ZR(U, ) =

Z R ( U i , ) U . . . U ZR(U, , ) where { Z ~ ( l i , , ) , . . . Z R ( U ; , ) ) is a jamily o j irredun- dant prime ideals that does not depend on the essential direct sum q,'uniform

modules chosen.

Proof. It remains to prove the uniqueness of the irredundant union and this follows from the well known fact that an ideal contained in a unior. of prime ideals must be contained in one of the ideals in the union. 0

Dually a module with finite dual Goldie dimension m has an onto map

with small kernel f : M --+ HI $ . . . @ H , where each Hi is hollow. By Lemma 4.7 W R ( n / I ) = W R ( H 1 ) U . . . U W , q ( H m ) and by Proposition 4.5, W R ( H i ) is a prime ideal of R. Thus we have:

4.9 Lemma. Let M be an R-module with finite dual Goldie dinension rn such that we have f: IM 4 H I $ . . . @ H,, and each H I is a h o l l o , ~ module.

Then there esists { i l , . . . , i k ) C { I , . . . , m ) such that W R ( M ) = I V R ( H ~ ) U

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. . . ~1 L C - I t ( H,,) = LC',y(H,, ) U . , . U CL-K(H,, j where { I .L 'R(H, , ). . . . : I .V,y(H, , )}

I S a f a r r ~ ~ l ~ of irrcdundan,t przrrlc i d c d s that does not depend o n the h o l l o 1 ~

modules chosen.

Of course the prime ideals appearing in the unions can also be maximal.

.As the nest Lemma shows this lvill h ~ p p e n if either 41 has non-zero socle or

a finitely generated quotient.

4.10 L e m m a . Let !\.I be a n R -modu le

( 1 ) If .\1 has nonzero socle t hen ZR(iLI) contazna a maxzma l zdeal If R raoetherzan and .lI has finztc Golclie dzmenszon t h e n the converae

zs also true

( 2 ) If the radzcal of M zs a proper submodule of -21, t hen bVR(ibl) contazns a maxzmal zdeal If R zs noetherzan and !?I has finzte dual Goldze

dzmenslon t h e n the converse zs also true

Prooj. (1) If :Lf has nonzero socle S then by Lemma 4.7 Z R ( S ) 5 Z R ( M ) ,

and since ZR(S) contains at least one maximal ideal also does ZR(n/f). If R is

noetherian, :VI has finite Goldie dimension and Z R ( M ) contains a maximal

ideal JM then by Lemma 4.8 iLf contains a uniform module Cr such that

ZR(U) = iM. Since by Lemma 4.7 we may choose U cyclic there exists an

element u of U with maximal annihilator and it follows that a n n ~ ( u ) = JM.

Thus uR is a simple submodule of M.

The proof of the first part of (2) is just dual to the argument for (1).

The second part holds because in a noetherian ring we also have coassoci-

ated primes, but we include the argument for completeness' sake: If R is

noetherian and M has finite dual Goldie dimension then by Lemma 4.9 1LI has a hollow quotient H such that kVR(H) = M . Since R is noetherian

JM = ( r , , . . . , r,)! since r , H C H and being H hollow any proper submodule

of H is small in H, we have that M H C H . Now H I M H is a nonzero hollow

R/JM-module, thus it is simple therefore hf has maximal submodules. 0

If M is a linearly compact module over a commutative ring R , then M has finite Goldie and dual Goldie dimension so Lemmas 4.8 and 4.9 can be

applied to M.

Assume that M is also indecomposable, if i M denotes the maximal idea!

of EndR(M) then it is clear that P = M i? ( R l annR(M)) = z R ( U ~ ) U . . . U

ZR(Un) U T / S / R ( H ~ ) U . . . U FVR(H,). Since this union of prime ideals is an

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LINEARLY COMPACT MODULES 1239

ideal, either P = Z R ( L i , ) for some 1 or P = I /VR(H,) for some i . So we ha\,e

proved:

4.11 Proposi t ion . Let !LI he n lznearly compact module over a rzng R

T h e n

( 1 ) Ih(,\l) and ZR( . l l ) are a finite unzon of p r z m e d e a l s of R ( 2 ) If zn addztzon -11 z s znd~composable t hen ezther WR(!LI) zs a przme

ideal and t hen ZR( . l I ) 5 l17R(ALI) or Z R ( 3 1 ) ta a przmP zi'eal and

t h ~ n WR( . l I ) C ZR( . l l )

According to Vamos [I.?. $3) we say that a ring R with a pr i~ne ideal P is a lexicographic extension of Rp by R I P provided that for any ic!eal I of

Reither I C P or P C I . Vamos proved [V2. Lemma 3.1 and Theorem 3.21 that if R is :.L lexico-

graphic extension of R p by RIP such that R L+ Rp, then R is 1.c. if and

only if R p is a 1.c. ring and Q ( R / P ) is 1.c. as R-module. An ana1ogo.m result

can be proved for ERC:

4.12 T h e o r e m . Let R be a 1.c. ring wi th a prime ideal P such that h + Rp,

and R is a lexicographic extenszon of R p b y R I P . T h e n ERC holds for R if

and only if i t holds for R p and R I P .

Proof. Assume that ERC holds for R then by [V2, Lemma 3.11 R p is 1.c. as R-module, thus by Proposition 1.15 R p and RIP satisfy ERC. To prove

the converse by Theorem 1.14 it is enough to show that HomR(:\,f,. M 2 ) is R-1.c. for Mi and 1% indecomposable R-1.c. modules. Let

and

Kz = { m E M z / P anng(m)).

For each i , Ki is an RIP-submodule of i l l i such that ZR(M,/h ' , ) P If we apply the functor Horn( -, M2) to the exact sequence

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1240 FAIT11 A N D I IERBERA

Since R I P satisfies ERC, we conclude that HomR(hf l , hf?) is 1.c. if and only

if HomR(hTi I I<, , ibf2 ) is. If' iV is an i~ldeconiposable su~rirnalid of illl / I C l then ZR(iV) P, without lost of generality we may assume that M1 is

an indecomposable R-module such that Z ~ ( l l / l l ) 2 P. A dual and similar

argument ~ v i l l yield that we may also assume that !Lf? is an indecomposable

R-module such that ZR(1b12) P .

In this situation if WR(ilil1 ) C P then iLT1 and

are Rp-mociules. Since Rp satisfies ERC,

is 1.c. as Rp-module and by [V2, Lemma 3.11 and Proposition 1.7 it is

also R-linearly compact. A similar argument will work if bVR(A.l?) C P,

since HomR(lbfl, hf i ) 2 H o m ~ ( h f ; , lv;), and ZR(lb12) = bvR(hf;) and also

\\'R(lbf2) = ZR(&f;).

Assume now that P 5 W~(&/12). Then WR(Pf;) 2 P $ ZR(M;). Let

Applying the functor H o m ~ ( - , Dl;) to the exact sequence

is 1.c. if and only if HomR(M,'/K, &I:) is.

But ZR(-\I;/K) P and since by Lemma 4.7 WR(!K/I<) C bp-R(!LI;) P, we can conclude that HomR(M1, M2) is 1.c. as R-module. O

An example of lexicographic extension is the following ([V2, Example 3.31):

Let Ii be any field and consider R = K [ [ x ] ] + yK((x))[[y, -.I] + zI<((x))[[y, z ] ] .

The ideal P = ~ K ( ( s ) ) [ [ y , z]] + z K ( ( z ) ) [ [ ~ , z ] ] is prime and any ideal of R is

either contained in P or contains P. R p = K((x)) [ [y , r]] and R I P = I<[[z]]

both are noetherian complete rings thus, as we will see in Theorem 6 .2 , they

satisfy ERC thus R also satisfies ERC. Notice that R is neither noetherian

nor uniserial.

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LINEARLY COMPACT MODULES

5 . LOOKING FOR A L I N E A R L Y COhlPr \CT RING

In this section we will show that if hf is a finitely embedded linearly

compact module or a uniserial linearly compact module then EndR(!M) is a

ring with right and left Morita self-duality (cf. Corollary 5.7 and Theorem

5.10).

5.1 Lemma. Let R be a rzng, E a cogenerator S = E n d R ( E ) a n d I f an

R-module Denote by .\I* the E-dual of !Ll then

(1) a n n R ( M ) = annR(trs(A1))

(2) If 11' ,\!f*, then trE(I<) C t r ~ ( . \ l )

(3) t rE (h f ) = t rE(hf*)

Proof. (1) It is clear that annR(il/l) C annR( t rE(M)) . Since E is a cclgenera- tor, &I embeds in a product of copies of E which implies that annR(trE(iii[)) C_

a n n ~ ( M ) . To prove (2 ) it is enough to show that if a: = u! f ) ~vhere f E: I< and

u E Homs(K, E ) then x E t r ~ ( M ) . If x $! trE(!\l), since E is a cogenerator, there exists g E S such that g( t rE(M)) = 0 but g(x) # 0. Thus gf = 0 and

g(x) # 0, but this is impossible because u is a S-homomorphism and we may

conclude that x E t r E ( M ) .

By (2) we have that t r E ( M * ) C trE(i\f). Let z E trE(&f), then x =

f l ( m l ) + . . . + f n ( m n ) , where f , E iW* and m, E ,bf. Denote tly urn E

Horns(h/lt, E) the evaluation map at m, that is for f E ibf* u,(f) = f ( m ) ,

then x = orn , ( f l ) + . . . + o,,,(f,) E t rE(AI*). O

A module M is said to be N-injective if every homomorphisn.1 from a

submodule of A' to M can be extended to an homomorphism from N to M .

Let R be a ring and E an injective cogenerator, a module ibI is linearly

compact if and only if M is E-reflexive and E is M*-injective, cf. [X, 'Theorem

4.11. The following Lemma shows that this characterization can be easily

rewritten in terms of t r ~ ( M ) .

5.2 Lemma. Let R be a ring and E an injective cogenerator. Then a module

M is linearly compact if and only if M is tr~(rLI)-rejexive and t r E ( M ) is

HomR(M, trs(M))-injective.

Proof. Since E is injective H o m ~ ( t r ~ ( A 4 ) . E) = E n d ~ ( t r ~ ( 1 b f ) ) is an onto

image of E n d R ( E ) . In fact EndR(trE(hf)) Z EndR(E)/ I . where I = { f E

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1242 FAITH A N D [HERBERA

E n d R ( E ) ( f(trE(iL/I)) = 0). Then the structure of E r id~(E) -modu le of

is in fact a structure of EndR(trE(lb!))-module. It is clear now that !LI is

E-reflexive if and only if it is trs(,\I)-reflexive. Let Ii C Ad* and let f : I< i E be an homomorphism, by Leinnia 5.1(2)

f(I<) C t r E ( M ) , thus E is .Lf'-injective if and only if trE(!L/) is !\I*- injective. D

5.3 L e m m a . (Varnos [ V l , Theorem 2 . l j ) Let R be a ring, 1bI a module with

essential simple socle and E ( h I ) its injective hull. If xR, T E )\I, is 1.c. then

for any f E E n d R ( E ( M ) ) , f (xR) C xR.

Proof. Assume that f E EndR(E(iZ/I)), then f ( x ) = y and a n n ~ ( . r ) C annR(y). The module N = {y E E ( M ) 1 a n n ~ ( x ) C a n n ~ ( y ) ) is an injective

hull of xR as R / annR(x)-module. ilie are assuming that sR 2 R/ annR(x)

is a 1. c, subdirectly irreducible ring, by dnh ' s result (cf. Theorem 1.5) it is

also a self-injective ring. Thus IV = xR and f ( x ) = y E xR.

If M is l.c., and Jve denote by E an injective cogenerator of R then

h/l is E-reflexive, or as we showed in Lemma 5.2 tr~(n/I)-reflexive, M E

Homs(M*, t rE (M)) , where 5' = E n d ~ ( t r ~ ( M ) ) . Hence has also a mod-

ule structure over the biendomorphism ring of t r ~ ( n / o . On the other hand we have tha t M has a module structure over the completion of R in the

M-topology, we will show that both rings are the same.

5.4 Lemma. Let M be a 1.c. module over a local ring R with minimal in- jective cogenerator E . Then the completion of R in the M-topology and i n the t r ~ ( M ) - t o p o l o g y coincide.

Proof. For any A = { m l , . . . ,m,) C M, R/ a n n ~ ( m 1 , . . . , m,) is 1.c. (cf. The- orem 1.10). By Lemma 5.1

R / a n n ~ ( m 1 , . . . . m,) = R / a n n ~ ( t r ~ ( m l R + . . . + m,R)),

hence if T denotes a finite subset of t r ~ ( r n 1 R + . . . + m,R) the natural

map 9 ~ : R/ a n n ~ ( A ) ---t R/ ann(T) is injective and because of linear

compactness is in fact bijective.

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LINEARLY COMPACT MODULES 1243

Now let T denote a f m t e subset of trg(.\Il. chen there exlsts d fin] te subset

A such that T t rE(AR) Denote 5) R the completion of ,'? In the

i\l-topolog> and by S the completion of R in the trE(hf)-topology The set

of projections

induce a map

9: R + s, since 9.4 are isomorphisn~s it is easily seen that p is also an isomorpi:.ism. G

5.5 P ropos i t i on . Let R be a local ring with minimal injectzve cogenerator

E and .\I a I. c, module over R, then:

(1) S = EndR(trE(h!f)) = Ends(trE(iV1)) is a commutative local rzng and the completion of R in the trE(M)-topology.

(2) S is complete in the IV-topology for any ilr C t r~(iV1).

(3) S zs the completion of R in the I?!-topology.

(4 ) S is a linearly compact ring if and only if trE(!b!) is 1.c. as S-module

if and only if trE(hLfj is linearly compact as R-module. Moreover in

this sit.uation trE(lV1) is the minimal injective cogenerator (.IS.

Prooj. (1) Since t r E ( M ) is quasi injective, S Z E ~ ~ R ( E ) / I where 1 = {f E

S I f ( I ) = 0). Since EndR(E) is local S is also local. M is 1.c. thus XR is 1.c. for any x E t rE(M), thus we may apply Lemma 5.3 and conclude

that S is commutative. Also by Lemma 5.3 Sx = xR, thus Sx is 1.c. as S- module, so again by Lemma 5.3, we may conclude that T = Endsl:tr~(l?!))

is commutative. Now sh'lT is a balanced bimoduie over commutative rings,

thus S = T. The fact that S is the completion in the tr~(ll/l)-topology of R follows from [Vl , Proposition 1.51.

To prove (2) observe that by Lemma 5.3 any submodule N of t rE (M) is

quasi-injective thus its endomorphism ring is a quotient of S and then we

can apply again [Vl , Proposition 1.51 to conclude that the endomorphism

ring of N is the completion of R in the N-topology so S is complete in the

!\'-topology.

Statement (3) follows from (1) and Lemma 3.4. Notice that since S is the completion of R in the trE(M)-topology, the

structure of trE(:LI) as S-module is the same as the structure as I?-module,

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1244 FAITH A N D I{ERBL'.RA

thus tss(.Llj is still a finitely eriibetided S-iiiotlulc ivi~!l slrliple s o c l e and

trE(.Vl) is R-linearl). compact if and only if it is S-linearly compact.

If S is linearly compact then by Theorem 1.5 its minimal injective cogen-

erstor is also linearly compact, since t rE (M) is contained in this rnininial in-

jective cogenerator it is also linearly compact, and since S = Ends(trE(ibl)),

trE(.\I) is in fact the minimal injective cogenerator of S. Xssurne that t r E ( M ) is 1.c. as R-module, then by Lemma 5.2 t r E ( M ) is

S-injective, that is injective as S-module and hence is the minimal injective

cogenerator of S. Thus S is a ring with a balanced injective cogenerator. so

by [S, Theorem 2.41 it is a ring with hforita self-duality or equivalently S is

a linearly compact ring as we wanted to see.

We do not know whether it is true in general that the trace of a 1.c. module

on a minimal injective cogenerator is l.c., but notice that the statement for

finitely generated modules is equivalent to Anh's result (Theorem 1.5) as we

will see in the next proposition. A positive answer to this question would

give a general way to associate a 1.c. ring to a LC. module, then if S satisfies

ERC we could conclude that the endomorphism ring of the linearly compact

module has Morita self-duality.

5.6 P r o p o s i t i o n . Let R be a local ring with minimal injective cogenerator

E . If ?d is a finitely generated or finitely embedded 1.c . module then t r E ( M )

is LC.

Proof. If M is finitely generated then R/ ann~(i2. l) ~t M n and as a conse-

quence the M-completion of R , R/ annR(!M) is a 1.c. ring, thus it is com-

plete in any linear topology. By Theorem 1.5 the minimal injective cogen-

erator of R / a n n R ( M ) is also linearly compact, and it is easy to see that

t r E ( M ) = {I E E I a n n ~ ( x ) > a n n ~ ( h ! ) ) is isomorphic the minimal injec- tive cogenerator of R/ annR(M), thus trE(A/I) is linearly compact.

If Ad is finitely embedded we have an injective homomorphism 6: hC v En. Let n;: En --+ E denote the natural projection on the i-th component. We

claim that t r E ( M ) = C ; n = l ~ , ~ ( M ) . Let f be a homomorphism from .2/1 to

E, this extends to a homomorphism f from $:=l~,~(?il) to E, by Lemma

5.3 f ( l r , ~ ( M ) ) 2 T,E(M) thus f ( M ) ', C:,,T,E(-Z/I) and the claim follows.

Since t r E ( M ) is a finite sum of 1.c. modules we can conclude that it is a

1.c. module.

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LINEARLY COMPACT MODULES 1245

5.7 Corol lary . Let R be a ri.ng and ILI a finztely embedded 1.c. R-module ,

(1) if i\/l has szmple socle, then End~( , \ . f ) as a comn~utatz~ue 1.c. l x a l ring

wi th minimal injective cogenerator M .

(2) EndR(.2/1) is a ring with Morita self-dualzty.

Prooj. By Lemma 1.3 we may assume that R is local. Let us deric.~te by E the injective hull of the unique simple module. If I ~ I has simple soclt. lve may assume that 1l.l C E, and by Lemma 5.3 = t rE(M) : hence statement (1)

follows from Proposition 5.5. In general if 1bI is a finitely embedded 1.c. module over a local ring R, by

Proposition 5.6 irE(i21) is 1.c. and S = E n d ~ ( t r ~ ( i l 1 ) ) is the co.mpletion of R in the 34-topology. By Proposition 5.5 S is a ring with Morita self- duality. Since M is also finitely embedded as an S-module by Theorem 1.12 EndR(hf) = Ends(M) is a ring with Morita self-duality.

It is well known that any injective endomorphism of a right artinian mod-

ule is bijective. The following result shows that, at least in the commutative

case, this can be extended to finitely embedded 1.c. modules.

5.8 Corol lary . Let R be a ring and M af ini te ly embedded 1.c. module. T h e n

any injective endomorphism of .V is bijective.

Proof. Let iVI be a finitely embedded 1.c. module and f an injective endo- morphism of M. As in the previous Corollary we may assume that R is a

commutative ring with Morita self-duality. thus f induces an onto tmdomor-

phism f * of Ad* which is a finitely generated module, thus f* and f are

bijective. 0

We now specialize our results for finitely embedded linearly compact mod- ules to artinian modules,

5.9 Corollary. Let R be a ring and h.1 a n artinian module,

(1) (Facchini [Fa, Theorem 2.8)) zj M has simple socle then EndR(iLI) is a commutative complete noetherzan ring with minimal injective

cogenerator iVf. ( 2 ) If M has Goldie dimension and dual Goldie dimension 1, then 0 is a

pr imary ideal of EndR(M).

( 3 ) (Ballet [Ba, page 3751) EndR(-bf) zs a noetherian ring with Morzta

self-duality.

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1246 FAITH A N D 11ERRERA

Prooj. (1) In vie\v of Coi.oilary 5.7 \ye only iiecd to pro\-(. that E ~ l c i ~ ~ ( . \ I j is

noetherian. But EndR!hrl) is a ring with Morita duality and :L.l is the minimal

injective cogenerator, thus the lattice of ideals of EndR(Xi) is antisomorphic

to the lattice of s~ibmodules of .2f [X, Theorem 2.61, hecce if .\'I is artinian

then EndR (:bi) is noetherian. If h/l has Goldie and dual Goldie dimension 1, then the prime ideal P =

{ f E E n d R ( M ) / f is not onto) = {f E End~(lL!f)/ f nilpotent) is a nilpotent ideal, thus 0 is a primary ideal of E n d R ( b l ) as we claimed in ( 2 ) .

In general if !If is an artinian module then t r ~ ( M ) is artinian (where E denotes the minimal injective cogenerator of R) . Thus S = EndR(trE(!bI)) is noetherian and Endn(Rf) = Ends(lVf) is noetherian.

If Id is an artinian module with simple socle then iLI = N1 + . . . + !V, where each N , has Goldie and dual Goldie dimension 1. Since by Lemma 5.3

h/f and )Vi are quasi-injective modules,

by Corollary 5.9 this is a primary decomposition of EndR(hI) .

The same kind of argument can be done with a finitely embedded linearl!. compact module with simple socle, again ild = iVi + . . . + X, where each ;'i,

has Goldie and dual Goldie dimension 1 and by Lemma 5.3

We know that EndR(!v,) is a commutative local ring, with Goldie dimension 1 and by Corollary 5.8 and Corollary 4.6 we have that P, = WEnd,(N~)(N1) .

is a prime ideal of EndR(Ni ) , while the maximal ideal is ZEndR(N;)(Ni) .

We will prove now that if M is a uniserial linearly compact modules then

its endomorphism ring is linearly compact. Our way of proving this result follows the idea we believe may be true in general. That is, first we prove that

the trace of such a module is linearly compact and then by Proposition 5 . 5 we may assume that M is a linearly compact module over a linearly compact

ring. In fact in this situation we show that the endomorphism ring of the

trace of 1M on the minimal injective cogenerator is a a maximal valuation ring. Since ERC holds for masimal valuation rings we have that EndR(;Li)

is a linearly compact ring.

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LINEARLY COMPACT MODULES 1247

5.10 T h e o r e m . Let ;\I and S be nozero untserzal 1.c. modules ouer a local

rillg R with m i n i m a l injective cogenerator E . Then:

(1) S = EndR(trE(1Z/I)) i$ a maz ima l v a h a t i o n ring wzth m in ima l injec-

t ive cogenerator t r ~ ( i b f ) . ('2) EndR(l\il) i s 1.c. ouer S. ( 3 ) M @ R N is uniser ia l and R- l inear ly compact.

Prooj. First we will proof that the trace of a uniserial linearly compect mod-

ule is a uniserial module. Consider x and y two nonzero elements in tr ,g(M),

since .Jf is uniserial z E t rE (mR) and y E t r ~ ( n R ) for certain m: n E. hf . 'CFre

may assume that m R C n R and thus x ) y E tr,g(nR), but nR 2 R/ s n n ~ ( n )

which is a 1.c. uniserial ring, thus is a maximal valuation ring and its minimal

injective cogenerator t r E ( n R ) is a 1.c. uniserial module, hence xR C: yR.

By Proposition 5.5, ,!? = EndR(trE(h.1)) is the completion of .? in the trs(M)-topology and it is a complete ring in the N-topology for any !V C trE(!l/l), using this fact we may proceed as in the proof of Proposition 3.1 (1) to conclude that S is uniserial.

By Proposition 5.5 S is the completion of R in the .Vf-topology: hence

:;If is a faithful 1.c. S-module, thus n,,, anns(m) = 0 and S/ an:ls(m) %

mS is linearly compact. Since S is uniserial any nontrivial ideal I of S contains anns(m) for a certain m E ICI thus S/I is linearly compact since

S/ anns(m) is. This shows that S is an almost maximal valuatjon ring, by [Vl, Proposition 4.41 the minimal injective cogenerator of S is linearly

compact as S module, hence t rE(M) is linearly compact as an S-module.

By Proposition 5.5 also S is LC., thus S is a maximal valuation ring and its

minimal injective cogenerator is t r ~ ( M ) . This finishes the proof of (1).

By Lemma 1.4 the statement (2) is a consequence of (1) and Theorem 3.3. By Proposition 3.1 M N is uniserial. Since t rE (M $ N) = t r ~ ( M ) +

trE(lV) by (1) i t is linearly compact, thus by Proposition 5.5 the completion

T of R in the M $ N-topology is linearly compact. Now by Lemma 1.4

L R ( M g R !V) = C ( M &- N), hence by Lemma 3.2 31 2~ :hr is linearly

compact as T-module and by Lemma 1.4 M @R N is linearly compact as

R-module. This proves statement (3). O

~ . N O E T H E R I A N RINGS

In this section we will specialize to linearly compact modules over noe-

therian rings. We will prove that E R C holds for noetherian rings and using

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1248 FAITH A N D HERBERA

tlie results of seccio~i 4 we rvill get also some inforniatioii (311 the structure of

linearly compact modules over noetherian rings.

First we prove that the tensor product of artinian modules over a commu-

tative ring has finite length. We believe that the result should be well known but we failed to find a reference for it in the literature.

6.1 Proposition. Let R be a ring and M and artinian R-modules , then

ilf @ R -Rj is a module of finite length.

Proof. (with A. Facchini) By Lemma 1.3 we can assume that R is local. Let

J denote the Jacobson radical of R , since N is artinian the chain M > M J > M J 2 . . . must terminate, so there exists n such that i\ilJn = ;WJn+k for

any k 2 0. Every element of N is annihilated by a power of the Jacobson

radical thus MJn @ N = 0 and M @R N 1 M / M J n @ R 11'. Similarly we can show that iVl @ R iV Z il/l/MJn @ :V/:VJm, were N J m =

N J " + ~ for any k 1 0. It is easily seen that the tensor product of modules

with finite length has also finite length, thus M R R iV iVf/iVf Jn 6 3 ~ N / N J m has finite length. 0

6 .2 T h e o r e m . Let R be a noetherian ring

( 1 ) If M is a linearly compact module then the completion of R i n the M-topology is a finite product of noetherian complete local rzngs.

(2) If M is a 1.c. module then EndR(hf) zs a ring with Morita seif-duality.

(3) If R is 1.c. t hen for any R-1.c. module EndR(M) is R-1.c.

( 4 ) If M and N are linearly compact modules then &l@Rf l is also linearly

compact.

Proof. Let R be a noetherian ring and denote by E its minimal injective cogenerator. To prove (1) by Proposition 5.5 it is enough to show that if M is R-1.c. then EndR(trE(M)) is a finite product of complete noetherian local rings. Since E is a direct sum of artinian modules by Lemma 1.3 the

trace in the minimal injective cogenerator of any 1.c. module is a finite sum of

artinian modules with essential simple socle. By Corollary 5.9 and Lemma 1.3 its endomorphism ring is a finite product of commutative complete local

noetherian rings. Thus statement (1) follows.

To prove statements (2) , (3) and ( 4 ) , by Lemma 1.3 we may assume that

R is local noetherian and by Lemma 1.4, Proposition 5.5 and (1) that it is

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LINEARLY COMPACT MODULES 1249

also complete. By Proposition 1.7 statement (2 ) will folloiv from (3; and by

Theorem 1.14 statement (4) will also follow from (3) .

To prove (3), by Theorem 1.14. it is enough to show that Homli(M: IV)

is 1.c. Lvhen ;\I is artinian and S is noetherian. This is very easy :.o prove

directly or since

HOmR(hI, !\;I r ( h I & R ~v* ) '

and N * is artinian, the result also follows from Proposition 6.1.

6.3 Proposi t ion . Let R be a noetherian 1.c. rzng and A!! a 1.c. ~ndecom-

,vosable faithful R-module with zero socle and such that J(ib1) = :\.I. Let jLi be the maximal zdeal of EndR(!bl). Then R is a local 1-dimensionai' Cohen-

Macaulay ring with minimal prime ideal P = R n iLi and R p is an artznzan ring which is LC. as R-module. I n particular 1LI viewed as Rp-module is a

module with finite length, ZR(h!!) = I/VR(*Z.l) = P and E n d ~ ( n / l ) is noether-

ian as Rp-module.

Proof. Since M is faithful, by Lemma 1.3 R is a local ring. By Corclllary 4.2 and Lemma 4.10 we know that R I P has Krull dimension 1. By Proposi-

tion 4.1 and Theorem 6.2 R p is 1.c. as R-module.

By Proposition 4.4 E R ( R / P ) is a 1.c. R-module, thus by Corollary 1.2

there exists a finitely generated R-submodule A of E = E R ( R / P ) such that E I A is artinian as R-module. Hence

is an artinian R-module. Since ARp is a finitely generated RE.-module

ARp C Soc,(E) and x e have that E/Soc,(E) is an artinian R-module. Assume that E/Soc,(E) # 0, since it is an artinian R-module, it is also

an artinian Rp-module and the simple Rp-module Q(R1P) L, E l Soc,(E).

Thus Q ( R / P ) is artinian as RIP-module, but this is impossible because R I P is not a field. Therefore E = Soc,(E) so E is an Rp-module of finite length.

Since R p is 1.c.. R p = E n d R ( E ) = EndR,(E) and hence R p is an artinian ring.

By Proposition 4 .1 the set of zero divisors of R is contained in P, which

implies that Soc(R) = 0 and R v Rp. Since R is one dimensional and R p is artinian. we have that R is Cohen-Macaulay and P is a nilpotent prime

ideal.

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1250 I:AITH A N D llER13E11.4

Since 1.c. modules over artinian rings are finitely generated we may con-

clude that .lI is an Rp-iilodule of finite length. Hence any injecti\.e or

onto endomorphism of hf is bijective SO Z , q ( M ) = W,q(XrI) = P, and EndR(hf ) = Endx, (:)I) is a noetherian Rp-module. E

We know by Proposition 3.9 and Corollary 4.2 that if P is the prime ideal

in Proposition 6.3, then the integral closure of R I P in its ring of quotients is

finitely generated as RIP-module, and Q ( R / P ) is a.f.g. (i.e. & ( R I P ) is not

finitely generated but every proper submodule is). We will show now that for

noetherian domains Q ( R / P ) is essentially the only kind of indecomposable

1inear.l~. compact nlodule .\J such that Soc(:Gf) = 0 and J(:II) = :bI.

From SSTeakley's results jcf. [W]) it follows that a.f.g. linearly compact

modules are important in relation with noetherian rings but, as the following

result shows. they play no role in other classes of rings.

6.4 Proposition. Let R be a ring and ibI a n o n zero linearly compact R-

module. Denote by R the completion of R i n the ibf-topology. Then the

following statements are equivalent:

(1) R is noetherian, Soc(.bf) = 0 , J ( M ) = M and E n d ~ ( h / l ) is a 1.c. do-

main.

( 2 ) R is a complete noetherian domain with Krull dimenszon one and

M z Q(R) . (3) iZ.I zs a n o n artinian almost finitely generated module.

( 4 ) annR(:CI) is a non mazimal prime ideal, R = R / annR(M) is a noe-

therian domain and 121 Q(R/ annR(M)) .

Moreover i n this situation the integral closure S of R / a n n R ( M ) i n its

ring of quotients is finitely generated over R/ annR(M) and S is a discrete valuation ring.

Proof.

(1) + (2) By Lemma 1.4 End~( f \ / l ) = E n d R ( M ) , and as R-module i\/l is

still a 1.c. module such that Soc(M) = 0 and J ( M ) = M . By Theorem 6.2

R is a complete noetherian domain and Endg(M) is R-linearly compact.

By Proposition 6.3 there exists a prime ideal P of R such that ~p is

artinian and since Rp ~f EndR(M) it is also a domain, so Rp is a field and

~p = Q( R ) . Thus P = 0 and by Proposition 6.3 we can conclude that R has

Krull dimension 1. Since .\I is an indecomposable Q(R)-module. .if r Q(R) .

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LINEARLY COMPACT M O D U L E S 1251

( 2 ) =+ (3) follows from Proposition 3.9 and (3) + ( 4 ) follows frorn \Veal--

le~. 's results [\\;, Proposition 1.1. Proposition 1.3 and Proposition 1 41.

(4) + (1) is clear.

The rest of the proposition follows from Proposition 3.9. 0

Following Zoschinger [Zo], for any R-module h l kve can consider the sub-

modules: L ( M ) defined as the sum of all artinian submodules of M , and

P(iL1) defined as the sum of all submodules iV of ibf such that J ( N ) = :V. We will use the following facts about L ( M ) and P(1b1) due to Zoscliinger:

6.5 L e m m a . iWis, 41.10 pg. 3561 Let M be a 1.c. R-rnod.de. Theri

(1) L(~bI/L(i\ l l)) = 0 and L ( M ) is artinian. ( 2 ) P ( M / P ( M ) ) = 0 thus M / P ( M ) is noetherian and J(P(I1.I)) = P(A1).

Zoschinger in [Zo] proved that any onto endomorphism of a linearly com-

pact module over a noetherian ring has artinian kernel and dually that any

injective endomorphism of a linearly compact module over a noetherian ring

has noetherian cokernel.

If -\I is a 1.c. module then either P ( M ) I L ( P ( M ) ) = 0 and then by

Lemma 6.5 2b.1 is an extension of an artinian by a noetherian module. or

IY = P ( ~ V I ) / L ( P ( h I ) ) # 0 and then Soc(iV) = 0 and J ( N ) = N .

If R is noetherian complete then by Proposition 6.3 N = !V1 $ . , . 3 .V, where each !Vi is a module of finite length over Rp, for a certain prime ideal

Pi of R such that RIP , has Krull dimension 1. It is well known that modules

of finite length satisfy that any injective or onto endomorphism is bijective.

We will show now that this can be extended to N, as a consequence we will

find a new proof of Zoschinger result.

6.6 Lemma. Let M be a linearly compact module over a complete noetherian ring R such that Soc(h/f) = 0 and J ( M ) = &I. Then any injectivf or onto

endomorphism of M is bijective.

Proof. By Proposition 6.3 there exists a finite set of different prirne ideals

of R, { P I , . . . , P,) say, such that M = ibf1 @ . . . 8 M,, each M, is an Rp, module of finite length and the I<rull dimension of RIP, is one.

LVe will prove first that for any R-module homomorphism f : Ad, i !LIj ( i # 1 ) has essential kernel. Let m 6 M, be such that f ( m ) # 0. Sir-ce mRp,

as Rp,-module has finite lenght: WR,, ( m R p j ) = Z R ~ , (mRp, ) ::= P,Rp, .

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1252 FAITH A N D I I E R B E K A

hence bVR(mRp,) = ZR(mRp, ) = PI. If nlRp, n I ie r f = 0. then by

Lenlma 1.7 P, = Z R ( f (mRp, ) ) C Z R ( M J ) = P,. which is impossible since

Rp, has I<rull dimension 1. Thus mRp, n Kerf + 0 and as a consequence

m R n Kerf # 0. Assume now that f is an injective endomorphism of .ll and consider

where xJ and E , denote the natural projection and the natural inclusion

respectively. Since for any i f j f,, has essential kernel and f is injective

then for each i , f,, is injective. Since M, has finite length as Rp, module, f , ,

is in fact bijective.

By Lemma 1.3 M is a finite sum of indecomposable modules with local

endomorphism ring, M = N1 $ . . . !Vk say. As before f induces maps

j,, : Ni --+ N,. Since f,, are bijective and iVk has local endomorphism ring

jkk are also bijective, which in turn implies that f is bijective.

To prove that any onto endomorphism f of 11.1 is bijective consider the dual

map f * : M* --+ M * . f * is injective and Lve will also have that Soc(ib1') = 0 and J ( M * ) = M'. Thus f' and f = f'* are bijective.

6.7 Corollary. Let M be a linearly compact module ouer a noetherian ring

R. Then

(1) If f E E n d R ( M ) is onto then Ker f is artznzan. (2 ) I f f E EndR(h") is injective then .bf/ Imf is noetherian.

Proof. By Theorem 6.2 and Lemma 1.4 we may take the completion in the

M-topology of R and we may assume that R is noetherian complete. Since

in this situation we will have a self-duality and the statement (2 ) is dual to

(1) it is enough to show (1).

Assume that f is an onto endomorphism of M and that Ker f # 0. Since

f (P (M)) C P(M), f induces an onto map 1: M/P(!VI) i M/P(M). But

by Lemma 6.5 M / P ( M ) is noetherian and thus j is bijective. Hence Ker f C P ( M ) and f: P(M) --+ P(M) is an onto map. By taking P(M) instead of

M we may assume without lost of generality that J(M) = M .

Since f ( L ( M ) ) C L ( M ) , f induces an onto map f : i U / L ( M ) --+ ~bf/L(&l).

By Lemma 6.5 and Lemma 6.6 f̂ is bijective, thus kerf C L,(!L.I) which is an

artinian module by Lemma 6.5. O

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LINEARLY COMPACT MODULES 1253

6.8 P ropos i t i on . Let h1 be a linearly compact module over a rzoetherzan

ring R , then:

(1) If Soc(M) = 0 then S = EndR(.\f) is a left noetherian ring.

(2 ) If J ( M ) = 1\/1 then S = EndR(!bf) is a right noetherian ring.

Proof. By Theorem 6.2 and Lemma 1.4 we may assume that R is the com-

pletion of R in the !)I-topology, so we may assume that R is complete and noetherian. Thus R will have a Morita duality and since the statement (2)

is dual to (1) it is enough to show (1) .

Apply the functor HomR(M. ) to the exact sequence

Since P(M/P(M)) = 0,

and being l V f / P ( M ) a noetherian R-module we have that End~(h"/P(!bf) )

is finitely generated as R-module.

HomR(hf, P ( M ) ) is a two-sided ideal of S, thus to prove that s,.? is noe-

therian it is enough to show that s HomR(M, P(.Zi()) is finitely generated.

Apply now HornR( , P( .V)) to (* ) . HomR(1\/l/P(h!!), P ( M ) ) is a left noe-

therian module over T = E n d ~ ( P ( . v ) ) = S/I, where I = { f E Sl f ( P ( h f ) ) =

0). Thus HomR(M, P(1Z.l)) = HomR(hd, P ( M ) ) is noetherian if and only

if T = E n d R ( P ( h I ) ) is left noetherian, but Soc (P(M)) = 0 thus by Proposi- tion 6.3 T is in fact a right and left noetherian ring.

If R is a complete noetherian ring and M a linearly compact R-module

EndR(M) may be neither right nor left noetherian. Take for example R =

K[[x]] and of = K[[x]] $ K[x-'1.

ACKKOWLEDGEMENTS

Part of this work was done while the second author was supported by a

postdoctoral fellowship from the Ministerio de Educaci6n y Ciencia of Spain

and the Fulbright Foreign Scholarship Board at the Mathematics Department

of Rutgers Unii-ersity. She wishes to thank her host for its hospita:ity.

The research of the second author was also partially supported by the

DGICYT (Spain), through the grant PB92-0586, and by the Conissionat

per Universitats i Recerca de la Generalitat de Catalunya.

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FAITH A N D HkKf3ERA

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Received: February 1996

Revised: November 1996

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