Modules with Indecomposable Decompositions That Complement Maximal Direct Summands

Transcription

1 Ž. JOURNAL OF ALGEBRA 197, ARTICLE NO. JA Modules wth Indecomposable Decompostons That Complement Maxmal Drect Summands Nguyen Vet Dung* Department of Mathematcs, Unersty of Glasgow, Glasgow, G12 8QW, Scotland, Unted Kngdom Communcated by Kent R. Fuller Receved September 13, INTRODUCTION Generalzng a fundamental property of semsmple modules, Anderson and Fuller 1 ntroduced the followng mportant concept for drect decompostons of modules. A decomposton M IM s sad to complement drect summands n case for each drect summand A of M, there s a subset H I such that M A Ž M. Žcf. also H 2, p Such a decomposton s necessarly an ndecomposable decomposton, and t s an nterestng problem to study under whch condtons an ndecomposable decomposton complements drect summands. There s an extensve lterature concernng ths problem. By 1, Theorem 6, rght perfect rngs can be characterzed by the property that all projectve rght modules have decompostons that complement drect summands. Fuller 7 showed that over a generalzed unseral rng every module has a decomposton that complements drect summands, thus provdng a frst class of non-semsmple rngs satsfyng ths property. More generally, Tachawa 14 proved that every Ž left and rght. module over a rng of fnte representaton type has a decomposton that complements drect summands, and that the converse s also true was establshed by Fuller and Reten 9. Restrctng just to one sde, Fuller 8 showed that rngs over whch every rght module has a decomposton that complements drect summands are precsely the rngs over whch every rght module s a drect sum of fntely generated modules Žthey are also called rght pure-semsm- * On leave from the Insttute of Mathematcs, P.O. Box 631, Bo Ho, Hano, Vetnam $25.00 Copyrght 1997 by Academc Press All rghts of reproducton n any form reserved.

2 450 NGUYEN VIET DUNG ple rngs, and t s stll an open problem whether rght pure-semsmple rngs are of fnte representaton type; see e.g., 11, 17.. Harada 10 developed the theory of factor categores nduced from a famly of modules wth local endomorphsm rngs, and used t to determne necessary and suffcent condtons for a drect sum of modules wth local endomorphsm rngs to complement drect summands. Note, however, that the local endomorphsm rng hypothess s not necessary for a decomposton to complement drect summands Žsee, e.g., 2, Exercses 12.5 and In ths paper, we remove all restrctve hypotheses and determne necessary and suffcent condtons for an arbtrary ndecomposable decomposton of a module, over any rng, to complement drect summands. Also accordng to Anderson and Fuller Ž 1 cf. 2, p. 141., a decomposton M IM s sad to complement maxmal drect summands f whenever M A X wth X an ndecomposable summand, there s an ndex I such that M A M. The sgnfcance of ths concept les n the fact that every decomposton of a module nto summands wth local endomorphsm rngs complements maxmal drect summands Žsee 2, Theorem Moreover, f M IM s an ndecomposable decompos- ton that complements maxmal drect summands, then the concluson of the KrullSchmdt Theorem holds true,.e., an ndecomposable decomposton of M s unque up to somorphsm 2, Theorem Obvously every decomposton that complements drect summands also complements maxmal drect summands. The man result of our paper s the followng. THEOREM. Let R be any rng and M a rght R-module wth an ndecomposable decomposton M IM that complements maxmal drect sum- mands. Then the followng condtons are equalent: Ž. a The decomposton M IM complements drect summands; Ž b. Eery non-zero drect summand of M contans an ndecomposable drect summand, and the famly M I4 s locally sem-t-nlpotent; Ž. c Eery local drect summand of M s also a drect summand. In the case n whch each summand M has a local endomorphsm rng, we redscover Harada s well-nown Theorem 10, Theorems and In contrast to Harada s categorcal proof, our method of proof s completely module-theoretc, and s nspred by some deas of Zmmermann-Husgen and Zmmermann 16 who showed n ther paper that the fnte exchange property mples the unrestrcted exchange property for modules wth ndecomposable decompostons. Our theorem s applcable also to decompostons n whch the ndecomposable summands need not have local endomorphsm rngs, as llustrated by two other consequences. Frst, we obtan a complete characterzaton of CS-modules whch have ndecomposable decompostons that complement

3 INDECOMPOSABLE DECOMPOSITIONS 451 maxmal drect summands. Ths generalzes recent results n 4, 5 on CS-modules whch are drect sums of modules wth local endomorphsm rngs. In partcular, we deduce that f M s a CS-module, and M has an ndecomposable decomposton M IM that complements maxmal drect summands, then ths decomposton complements drect summands. The fnal consequence, whch gves necessary and suffcent condtons for a drect sum of hollow modules to be quas-dscrete, substantally mproves Mohamed and Muller 12, Theorem 4.48 and thus provdes a more satsfactory answer to ther open queston n 12, p PRELIMINARIES Throughout ths paper we consder assocatve rngs wth the dentty and untary rght modules. An Ž nternal. drect sum I A of submodules of a module M s called a local drect summand of M f F A s a drect summand of M for any fnte subset F I. If, furthermore, I A s a drect summand of M, then we say that the local drect summand I A s also a drect summand of M. A famly of modules M I4 s called locally sem-t- nlpotent f for any nfntely countable set of non-somorphsms f n : M n M 4 wth all n dstnct n I, and for any x M, there exsts a n1 1 postve nteger Ž dependng on x. such that f... f Ž x Followng Crawley and Jonsson 3, a module M s sad to have the Ž fnte. exchange property f, for any Ž fnte. ndex set I, whenever M N A for modules N and A, there are submodules B of A Ž I. I such that M N M Ž B. I. A module A s called ndecomposable f t s non-zero and cannot be expressed as a drect sum of two non-zero submodules. A drect summand K of a module M s called a maxmal drect summand f M K X wth X ndecomposable. Recall that a decomposton M IM s sad to complement Ž maxmal. drect summands n case for each Ž maxmal. drect summand A of M, there s a subset H I such that M A Ž M. H. We wll refer to Anderson and Fuller 2 for all undefned notons used n the text, and also for basc facts concernng ndecomposable decompostons of modules. For the reader s convenence, we record here some of the nown results whch wll be used repeatedly n the sequel. LEMMA 2.1 Žsee 15, Proposton 1.. An ndecomposable module M has the exchange property f and only f EndŽ M. s local. LEMMA 2.2 Žsee 3, p Let M be a module wth the exchange property, and suppose that A M B E Ž A. E for some I

4 452 NGUYEN VIET DUNG modules A, B, E, and A. Then there are submodules C of A such that AMŽ C. I E. LEMMA 2.3. Let M IM be an ndecomposable decomposton that complements maxmal drect summands. If M appears at least twce n ths decomposton Ž.e., there s an ndex j such that M M. j, then EndŽ M. s a local rng. Proof. Ths was proved n 1, Proposton 4 Žsee also 2, Proposton LEMMA 2.4. Let M IM be an ndecomposable decomposton that complements maxmal drect summands. Let M j JNj be another ndecomposable decomposton of M. Then cardž I. cardž J. and there s a bjecton : I J such that M N Ž I. for all I. Consequently, the decomposton M j JNj also complements maxmal drect summands. Proof. Ths was proved n 1, Theorem 2, Corollary 3 Žsee also 2, Theorem 12.4, Corollary LEMMA 2.5 Žsee 2, Proposton Let M A B be a decomposton wth the correspondng projecton : M B. Let C be an arbtrary submodule of M. Then M A C f and only f the restrcton of to C s an somorphsm C B. 3. THE MAIN RESULTS In ths secton we prove our man theorem whch gves necessary and suffcent condtons for an ndecomposable decomposton to complement drect summands. The crucal part of the proof wll be based on our study of local drect summands n an ndecomposable decomposton that complements maxmal drect summands. We start wth the followng lemma whch s a frst step n ths study. LEMMA 3.1. Let M IM be an ndecomposable decomposton that complements maxmal drect summands. Suppose that M N1 N D, where each N Ž 1 m. m s an ndecomposable summand of M. Then there exst 1,...,m I such that M N for all 1 m and MN N Ž M.. 1 m I 1,...,m4 Proof. We use nducton on m. If m1, then M N1 D, so Ds a maxmal drect summand of M. Hence there s an ndex 1 I such that M M D. It follows that D I 4M, so we can wrte D 1 1 D, wth D M, I I 4 I 1. By Lemma 2.4, M N1 Ž D. I s agan an ndecomposable decomposton that complements maxmal drect summands. Snce M s a maxmal drect summand I

5 INDECOMPOSABLE DECOMPOSITIONS 453 of M, ether M Ž M. I N1 and we are done, or there s some ndex j I such that M Ž M. I D j. If the latter case holds, we get that Dj M, and snce M N1 D M D, we have M N 1, so t follows that Dj N 1. Thus N1 appears at least twce n the decompo- ston M N Ž D., so by Lemma 2.3, EndŽ N. 1 I 1 s a local rng. Therefore N1 has the exchange property by Lemma 2.1, hence there are submodules B of M Ž I. such that M N Ž B.. Each B s a 1 I drect summand of the ndecomposable module M, so ether B 0or BM. Ths, together wth the fact that N1 s ndecomposable, yelds that MN Ž M. for some t I, as requred. 1 t Now suppose that M N N D wth all N Ž 1 n. 1 n ndecomposable, and that the concluson of the lemma s true for all m n 1. By the nductve hypothess, there are ndces 1,...,n1 n I such that M N for all 1 n 1 and ž / MN N M, Ž. 1 n1 I where I I,..., 4 1 n1. By Lemma 2.4, the ndecomposable decomposton Ž. complements maxmal drect summands. Hence applyng the nductve step m 1 proved above for N and the decomposton Ž. n,we get ether M N1 Nn1 Nn ž I j4m/ for some j I, n whch case we are done Ž clearly then N M. n j,or there s some postve nteger wth 1 n 1 such that M N1 N1 N1 Nn1 Nn ž I M /. In ths latter case, we would get Nn N. From the equalty M N1 Nn D, t follows easly that D has an ndecomposable decompo- ston Žsee, e.g., 2, Lemma 12.2., so Nn appears at least twce n an ndecomposable decomposton that complements maxmal drect summands. One more applcaton of Lemma 2.3 gves us that EndŽ N. n s local. Therefore Nn has the exchange property by Lemma 2.1. Now t follows from Lemma 2.2 that ž / N N N N M Ž. 1 n1 n J for a subset J I. Comparng Ž. wth Ž., t s easy to see that JI4 t for some t I and N M, whch completes our nducton. n t

6 454 NGUYEN VIET DUNG The next result, whch s of ndependent nterest, wll be crucal for the proof of our man theorem. The proof below s nspred by some deas n the proof of Zmmermann-Husgen and Zmmermann 16, Theorem 5. THEOREM 3.2. Let M IM be an ndecomposable decomposton that complements maxmal drect summands. Suppose that N A4 s a locally sem-t-nlpotent famly of ndecomposable drect summands N of M such that A N forms a local drect summand of M. Then there exsts a Ž. Ž. subset H I such that M N M. A H Proof. If A s a fnte set, the result follows mmedately from Lemma 3.1. Therefore, from now on, we assume that A s an nfnte set. We proceed n two steps. Step 1. We suppose that N N for all, A. Consder the followng subset J of I: J j I M N, A 4 j. By Lemma 3.1, for each postve nteger n 1 and dstnct ndces 1,...,n n A, there are 1,...,n n I such that M N for 1,...,n. In partcular, ths mples that cardž J. n. As ths s true for every n 1, J s an nfnte set. Snce the famly N A4 s locally sem-t-nlpotent, t follows that the famly M j J4 j s also locally sem-t-nlpotent. Set J I J. Note that Ž N. Ž M. A J forms a local drect summand of M. Indeed, for any fnte subset F,..., 4 1 n A, by Lemma 3.1 there are,..., n J such that M N for 1, 2,..., n, and 1 n Ž. ž / M N N M, 1 n I 1,...,n4 so n partcular, N N Ž M. J s a drect summand of M. 1 n By Zorn s Lemma, there exsts a subset H I whch s maxmal Žunder ncluson. wth respect to the propertes that J H and Ž N. A Ž M. H forms a local drect summand of M. We show now that MŽ N. Ž M. A H. Suppose, on the contrary, that M Ž N. Ž M. A H. There are an ndex 1 I H and an element x M such that x Ž N. Ž M. 1 1 A H. Snce J H, t fol- 1 lows that 1 J. By the maxmalty of H, there are fnte subsets A1 A and H H so that for the fnte subsum K Ž N. Ž M. 1 A1 H1 of Ž N. Ž M. A H, ether M K 0 or M K s not a 1 1 drect summand of M. Then K, beng a fnte subsum of a local drect summand, s a drect summand of M. But K s a fnte drect sum of ndecomposable summands of M, thus by Lemma 3.1 there s a subset II such that M K Ž M. I. For I, let p denote the canoncal projecton M M correspond- ng to the decomposton M K Ž M.. By the choce of x, there I 1

7 INDECOMPOSABLE DECOMPOSITIONS 455 s I such that 2 p Ž x1. ž a AN/ ž HM /. 2 Let f 1: M M be the restrcton of the projecton p on M.If f1 s an somorphsm, t would follow by Lemma 2.5 that M K M 1 Ž M. I 4, whch s a contradcton to our choce of K. Hence f1 s 2 not an somorphsm. Note that M Ž N. Ž M. A H, so ths 2 mples that 2 I H J. Now we repeat the above argument wth x f Ž x nstead of x 1. Snce MMj for all and j n J, ths process would yeld the exstence of an nfnte sequence of non-somorphsms f : M M Ž n nj, n n n1 1, 2,.... such that f... f Ž x. n for all n 1. But ths s a contradcton to the locally sem-t-nlpotency of the famly M j J 4 j. Therefore, we Ž. Ž. get that M N M. A H Step 2. We now consder the general case,.e., let N A4 be a locally sem-t-nlpotent famly of ndecomposable drect summands of M such that A N s a local drect summand of M. Choose a set of representatves N B4 of the somorphsm classes of the N, A. For a fxed N we collect all those N whch are somorphc to N, and denote ther drect sum by L. Then we have N L. A B For each B, by Step 1, there s a subset J I such that ML Ž M. J. By Lemma 2.4, ths decomposton complements maxmal drect summands, and so each ndecomposable decomposton of L complements maxmal drect summands Žsee, e.g., 2, Lemma In partcular, t follows that any ndecomposable drect summand of L s somorphc to N. Therefore, for n B, L and L do not contan somorphc ndecomposable drect summands. Now consder the set T of all pars Ž B, I. wth B B and I I, satsfyng the property that M Ž L. Ž M. B I I. Defne n T the followng partal orderng: Ž B, I. Ž B, I f and only f B1 B2 and I1 I 2. In order to prove the theorem, t suffces to show that T has a maxmal element wth respect to ths orderng. In fact, suppose that there exsts a maxmal element Ž B, I. n T. We have then 0 0 ž B0 / ž II0 / M L M. If we rewrte the above decomposton, representng each L as a drect sum of copes of N, we obtan an ndecomposable decomposton that, by Lemma 2.4, complements maxmal drect summands. Suppose that B0 B. Then we can choose some B B. Now applyng Step 1 to L and 1 0 1

8 456 NGUYEN VIET DUNG ths ndecomposable decomposton, bearng n mnd the fact that L 1 and L have no somorphc ndecomposable drect summands for any B 0, t follows that there s a subset J0 of I I0 such that ž B0 / 1 ž IŽI0 J 0. / M L L M. Settng B B 4 and I I J, we have Ž B, I. Ž B, I and Ž B, I. Ž B, I., whch s a contradcton to the maxmalty of Ž B, I Hence B0 B, and we are done. Therefore, n order to complete the proof, t suffces to show that any chan n T has an upper bound n T, whch would allow us to apply Zorn s Lemma to deduce that T has a maxmal element. Let Ž B, I. K 4 be any Ž nfnte. chan n T,.e., for any, l K, ether B Bl and I I l,orbb l and Il I. Set B* KB and I* I. We wll show now that Ž B*, I*. K T whch would mply that Ž B*, I*. s an upper bound of Ž B, I. K 4. Ths means that we have to show the equalty Let us denote by M ž B* L / ž I I* M /. f : L M B* I* the restrcton on B* L of the natural projecton ž / p: M ž I* M/ II* M I* M. By Lemma 2.5, to prove that the above equalty holds s equvalent to showng that f must be an somorphsm. Tae any x KerŽ f.. Then x Ž L. Ž M. B* I I*. There s an ndex K such that x L, and snce B ž B / ž I I / M L M, t follows that ž / ž / L M 0, B I I* whch mples that x 0. Therefore f s a monomorphsm.

9 INDECOMPOSABLE DECOMPOSITIONS 457 We show now that f s an epmorphsm. Suppose, on the contrary, that f s not surjectve,.e., that fž L. B* I* M. Then there exst some I Ž K. 1 and some element x1 I M such that x1 1 1 f Ž L.. Snce Ž B, I. T, we have B* 1 1 ž B / ž II / 1 1 Let M L M. ž / ž / p : M M M M 1 I II I be the natural projecton, and let f : L M 1 B I 1 1 denote the restrcton of p1 on B L. Then f1 s an somorphsm by 1 1 Lemma 2.5. Set y f Ž x.. Snce I I* I, we can wrte y x x z, where x M and z M. Thus, there exsts I*I 1 II* 1 some I Ž 2 K. such that I I and x2 I I M. Note that x2 s the mage of y1 under the natural projecton ž / ž / ž / h : M M M M 1 I I I II M, I I Ž. Ž. 1 so we have x2 h1f1 x1 g1 x 1, where g1 h1f 1 : I M 1 I I M. Also, clearly x1 x2 f Ž y 1., and snce x1 f Ž B* L., 2 1 t follows that x fž L. 2 B*. Now we can repeat the above argument for x2 nstead of x 1. Smlarly, there s some I Ž K. 3 wth I I and an element x M, so that x fž L. and x g Ž x. I I 3 B* for some homo- 3 2 morphsm g : M M. 2 I I I I A standard nductve argument yelds an nfnte ascendng sequence I I I 1 2 n Ž K. wth non-zero elements x M for n 1 Žputtng n n I I n n1

10 458 NGUYEN VIET DUNG I 0., and a famly of homomorphsms g : M M, n I I I I n n1 n1 n Ž. such that x g g... g x for all n 1. Note that snce n1 n n1 1 1 ž B / ž II / ž B / n1 n1 n ž II n / M L M L t follows that L M. B B I I n n1 n n1 M, By Kong s Graph Theorem Žsee, e.g., Osofsy 13, Lemma 10., the locally sem-t-nlpotency of the famly N A4 mples the locally sem-tnlpotency of the famly L n N 4 B B. Therefore the famly n n1 M nn4 I I s also locally sem-t-nlpotent. Moreover, snce n n1 L and L do not contan somorphc ndecomposable drect summands for any n B, the modules M Ž nn. I I are parwse n n1 non-somorphc, hence all of the maps gn are non-somorphsms. Ths gves us the contradcton whch completes our proof. LEMMA 3.3. Let M IM be an ndecomposable decomposton that complements maxmal drect summands, and suppose that M I4 s a locally sem-t-nlpotent famly. If N A4 s any famly of ndecomposable drect summands N of M such that A N forms a local drect summand of M, then N A4 s also a locally sem-t-nlpotent famly. Proof. In order to prove the lemma, t suffces to show that, for any countable subfamly N,...,N,... Ž A., there are dstnct ndces 1 1, 2,...,,... n I such that M N for 1, 2,.... Snce the decomposton M IM complements maxmal drect summands, each N Ž A. s somorphc to some M Ž ji. j. Therefore clearly t s suffcent to consder the case n whch N N for all, l 1. By l Lemma 3.1, there must exst an nfnte number of ndces I such that M N, and the result follows. We are now ready to prove the man theorem of ths paper whch establshes necessary and suffcent condtons for an ndecomposable decomposton to complement drect summands. Indeed such a decomposton always complements maxmal drect summands Žbut the converse fals,

11 INDECOMPOSABLE DECOMPOSITIONS 459 n general: By 1, Theorems 5 and 6, any nfntely generated free rght module over a semperfect rng whch s not rght perfect provdes such an example.. THEOREM 3.4. Let R be any rng and M a rght R-module wth an ndecomposable decomposton M IM that complements maxmal drect summands. Then the followng condtons are equalent: Ž. a The decomposton M IM complements drect summands; Ž b. Eery non-zero drect summand of M contans an ndecomposable drect summand, and the famly M I4 s locally sem-t-nlpotent; Ž. c Eery local drect summand of M s also a drect summand. Proof. Ž. a Ž. b. Suppose that Ž. a s satsfed. Then clearly every nonzero drect summand of M contans an ndecomposable drect summand. The fact that the famly M I4 s locally sem-t-nlpotent was proved n 12, Theorem ŽIn 12, p. 30 Mohamed and Muller gave credt to Kasch and Zollner for ths result.. Ž b. Ž a.. Suppose that Ž b. s satsfed and let D be any non-zero drect summand of M. By hypothess, D contans an ndecomposable drect summand. By Zorn s Lemma, D contans a local drect summand A N maxmal wth respect to the property that each N s ndecomposable. By Lemma 3.3, the famly N A4 s locally sem-t-nlpotent, so t follows from Theorem 3.2 that M Ž N. Ž M. A H for some subset HI. In partcular, A N s a drect summand of D, so that DŽ N. A L for some drect summand L of D, hence of M. If L0, then by hypothess, L contans an ndecomposable drect summand L,soŽ N. A Ls agan a local drect summand n D, a contradcton to the maxmalty of the local drect summand A N. Therefore L0 whch mples that D A N, hence M D Ž HM.. Ths shows that the decomposton M IM complements drect summands. Ž b. Ž c.. Suppose that Ž b. s satsfed and let A N be any local drect summand of M. Snce each N s a drect summand of M, t follows from the mplcaton ŽŽ b. Ž a.. proved above that N has an ndecompos- able decomposton N N. Then N s agan a local j J j A, j J j j drect summand of M, and because each N s ndecomposable, t follows by Lemma 3.3 that N A, j J 4 s a locally sem-t-nlpotent j famly. Therefore, by Theorem 3.2, we get that N N A, j J j A s a drect summand of M. Ž. cb Ž.. Assume that Ž. c s satsfed,.e., every local drect summand of M s a drect summand. It s obvous that every drect summand D of M

12 460 NGUYEN VIET DUNG also satsfes the same property, hence by 12, Theorem 2.17, D has an ndecomposable decomposton. Therefore, n partcular, every non-zero drect summand of M contans an ndecomposable drect summand. We have to show now that the famly M I4 s locally sem-t-nlpo- tent. Consder any nfnte sequence 1, 2,... of dstnct elements of I and a sequence of non-somorphsms f f f 1 2 n M M M n By composng maps together and rendexng, f necessary, we may assume, wthout loss of generalty, that ether all fn are monomorphsms, or none of the f s a monomorphsm. Also, we may assume that I n N 4 n n, and wrte n for smplcty. Set M x f Ž x. xm 4 n n n n. Then t s n easy to chec that the sum Ý M s drect and Ž M. M n1 n 1 n1 n1 M for each n 1. Hence M s a local drect summand of 1 n1 n M. Set M* M n1 n. By hypothess, M* s a drect summand of M, so that M M* K for some drect summand K of M. We now proceed to show that K 0. Suppose, on the contrary, that K 0. As s shown above, K contans an ndecomposable drect summand X, and so we have K X L for some L K. Then M* L s a maxmal drect summand of M, so MM* LM for some t N. If L0, then agan L Y L for some t ndecomposable Y and a submodule L of L. Smlarly, M* L Mt s a maxmal drect summand of M, hence M M* L M M for some n N. We can assume, wthout loss of generalty, that t n. Then we can wrte the above equalty as ž / M ž n1 M M M 1 / n ž n / Mt L ž n M M 1 / ž n / MtL, t n whch gves us a contradcton because t n and ths s a drect sum. Therefore L 0, hence M M* M t. As we remared earler, t s suffcent to consder two cases. Ž. 1 Suppose that all fn are non-monomorphsms. Then, n partcular, f s not a monomorphsm, mplyng that M M t t t 0, whch s a contra- dcton because by the above equalty we obtan M M M Ž t t t 1 M. 0.

13 INDECOMPOSABLE DECOMPOSITIONS 461 Ž. 2 Suppose that all fn are monomorphsms. We can wrte, by the above, Any x M M M* M ž t1 M / M ž M t 1 t t / ž t M M 1 / ž t /. t1 can be represented as Ž. Ž. x y x f Ž x. x f Ž x. t t t m m m for some y t 1 M, x M wth t m, and m a postve nteger. Snce all f are njectve maps, t follows that y x x x 0 t t1 m and x f Ž x. t t. It follows that ft s surjectve, hence an somorphsm, a contradcton to our assumpton that all fn are non-somorphsms. Therefore we have shown that K 0,.e., M M* M 1. Now tae any element x M 1. There are a postve nteger n and elements x M Ž 1,...,n. such that Ž. Ž. x x f Ž x. x f Ž x n n n It follows that x x, f Ž x. x for 1,...,n1, and f Ž x. 1 1 n n 0, so we obtan that f f... f Ž x. 0. Ths shows that the famly n n1 1 M I4 s locally sem-t-nlpotent. As a consequence of Theorem 3.4, we deduce the followng result due to Harada 10, Theorems and whch s an mportant supplement to the classcal KrullSchmdtAzumaya Theorem 2, Theorem However, as was remared n the Introducton, Harada s proof reles heavly on hs theory of factor categores and does not seem easly accessble. COROLLARY 3.5. Let R be any rng and M a rght R-module whch s a drect sum M IM of modules M wth local endomorphsm rngs. Then the followng condtons are equalent: Ž. a The decomposton M IM complements drect summands; Ž b. The famly M I4 s locally sem-t-nlpotent; Ž. c Eery local drect summand of M s also a drect summand. Proof. Snce each M has a local endomorphsm rng, by 2, Theorem 12.6, the decomposton M M complements maxmal drect sum I

14 462 NGUYEN VIET DUNG mands, and every non-zero drect summand of M has an ndecomposable drect summand. Now the result follows mmedately from Theorem APPLICATIONS: CS-MODULES AND QUASI-DISCRETE MODULES In ths secton we apply our man theorem Ž Theorem 3.4. to study ndecomposable decompostons of CS-modules and quas-dscrete modules. In these stuatons the ndecomposable drect summands need not have local endomorphsm rngs. A module M s called a CS-module Žor extendng module. 6 f every submodule of M s essental n a drect summand. A module M s called quas-contnuous Žsee 12. provded M s a CS-module and whenever K and L are drect summands of M wth K L 0, K L s also a drect summand of M. Clearly CS-modules and quas-contnuous modules generalze quas-njectve and njectve modules. Note that quas-contnuous modules wth ndecomposable decompostons are farly well understood. By 12, Theorem 2.22 such decompostons always complement drect summands, and every local drect summand s also a drect summand Žthe proof of ths latter fact n 12 uses specfc propertes of quas-contnuous modules.. However, t s stll an open queston to characterze CS-modules whch admt ndecomposable decompostons Žcf. 12, Open problem 8, p Wth the help of Theorem 3.4, we are now able to gve a complete characterzaton of CS-modules whch have an ndecomposable decomposton that complements maxmal drect summands. We wll show that, n fact, such a decomposton complements drect summands. We start wth the followng elementary lemma. Recall that a module A s unform f any two non-zero submodules of A have non-zero ntersecton. A submodule C of a module M s called a complement submodule of M f C has no proper essental extensons n M. Obvously any drect summand of M s a complement submodule of M. LEMMA 4.1. Let M M be a drect sum of unform modules M, I and suppose that eery unform submodule of M s essental n a drect summand of M. Then eery non-zero complement submodule of M contans a non-zero unform drect summand. Proof. Let C be a non-zero complement submodule of M. Note that any cyclc submodule of M has fnte unform dmenson; thus C must contan a non-zero unform submodule U. There s a maxmal essental

15 INDECOMPOSABLE DECOMPOSITIONS 463 extenson V of U n C. Then V s agan unform, and furthermore a complement submodule of M Žcf. 6, p. 6.. By hypothess, V s a drect summand of M. Followng Mohamed and Muller 12, p. 4, a famly M I4 of rght R-modules s sad to satsfy Ž A. 2 f the followng chan condton holds: for any choce of x M, wth dstnct I, such that r Ž x. r Ž y. n n n n1 R n R Ž. for some y M jj, the ascendng sequence r Ž x. Ž nn. j n R becomes statonary. For modules M and N, we say that N s nearly M-njecte provded for any non-monomorphsm f: A N, where A s a submodule of M, f can be extended to a homomorphsm g: M N Žcf. 6, p Our next lemma generalzes 5, Lemma 2.3, where the unform drect summands M were assumed to have local endomorphsm rngs. Due to the proxmty of the arguments, we prove only the drecton ŽŽ a. Ž b.. whch wll be used n the proof of Lemma 4.3. Note also that Lemma 3.1 of the precedng secton s essental for the proof of ŽŽ a. Ž b.. we gve below. LEMMA 4.2. Let M IM be a drect sum of unform submodules, and suppose that ths decomposton complements maxmal drect summands. Then the followng condtons are equalent: Ž. a Eery unform submodule of M s essental n a drect summand of M. Ž b. M M s a CS-module for each par jni,and M I4 j satsfes Ž A.. 2 Proof. Ž. a Ž. b. Suppose that Ž. a holds. Then t follows easly that every fnte subsum of M IM s a CS-module, n partcular M Mj s a CS-module for each par j n I. To show that M I4 satsfes Ž A., by the proof of 4, Lemma 3.2 2, t suffces to prove that jm s nearly Mj-njectve for each j I. Now tae any j I, a submodule AM and a non-monomorphsm f: A M. Set A* x fž x. j j xa 4. Then A* A, and by hypothess A* s essental n a drect summand D of M. Clearly D s also unform, so by Lemma 3.1 there s an ndex I such that M D Ž M.. If j, t follows that A*MjDMj0, hence f s a monomorphsm, a contradcton. Thus j, so that M D Ž M. j. Let p be the natural projecton DŽ M. j jm. Then obvously the restrcton f *of pon Mj extends f. Therefore jm s nearly Mj-njectve for each j I. Ž b. Ž a.. The proof s essentally analogous to the proof of 5, Lemma 2.3 ŽŽ b. Ž a.., and so we omt t.

16 464 NGUYEN VIET DUNG LEMMA 4.3. Let M IM be an ndecomposable decomposton that complements maxmal drect summands. If M s a CS-module, then the famly M I4 s locally sem-t-nlpotent. Proof. Consder any nfnte sequence of non-somorphsms f n f f f 1 2 n M M M, 1 2 n wth dstnct I. For smplcty, we may assume that I n N 4 n n, and wrte n. Smlarly as n the proof of Theorem 3.4ŽŽ c. Ž b.. n, we may assume, wthout loss of generalty, that ether all fn are monomor- phsms, or none of the fn s a monomorphsm. Suppose frst that all f are monomorphsms. Let us denote x f Ž x. n n xm 4 n by Mn and set M* n1m n. Snce M s a CS-module, M* s essental n a drect summand D of M, so we can wrte M D C for some C M. IfC0, then t follows from Lemma 4.1 that C L L for a non-zero unform drect summand L and a submodule L C. Snce D L s a maxmal drect summand of M, there s I such that M D L M. Agan as n the proof of Theorem 3.4ŽŽ c. Ž b.., M s a local drect summand of M, so n partcular M n1 n s a drect summand of M, hence of D. It follows that M M s a drect summand of M. But M M s essental n M M,so MM M 1 M 1. Ths mples that f s an epmorphsm, hence an somorphsm, a contradcton to the hypothess. Therefore C 0, hence M D,.e., M* M s essental n M. Tae any non-zero element x M n1 n 1 M*. Then x x1 f1ž x1. xn fnž x n., where x M,1n. It follows easly that x x and f... f Ž x. 1 n 1 0, whch s a contradcton because all f Ž 1 n. are monomorphsms. Suppose now that all fn are non-monomorphsms. Tae any non-zero element x M, and set x f... f Ž x. Ž for n n n By Lemma 4.2 ŽŽ a. Ž b.., the famly M n N4 satsfes Ž A. n 2, so the ascendng se- quence rrž x1. rrž x2. rrž xn. must become statonary. Therefore there s a postve nteger m such that r Ž x. r Ž x. r Žf Ž x.. R m R m1 R m m.if xmr0, then because Mm s unform, t would follow that fm s a monomorphsm, a contradcton. There- fore we have x 0, mplyng that x f... f Ž x. 0. m m m1 1 1 Wth these preparatons, we can now use Theorem 3.4 to prove the followng result whch generalzes 5, Theorem 2.4.

17 INDECOMPOSABLE DECOMPOSITIONS 465 THEOREM 4.4. Let M IM be a drect sum of unform submod- ules, and suppose that ths decomposton complements maxmal drect summands. Then the followng condtons are equalent: Ž. a MsaCS-module; Ž b. M s a CS-module for any countable subset H I; H Ž. c M M s a CS-module for each par j n I, M I4 j satsfes Ž A., and M I4 s locally sem-t-nlpotent. 2 Furthermore, f M satsfes any of the aboe equalent condtons, then the decomposton M IM complements drect summands, and any local drect summand of M s also a drect summand. Proof. Ž. a Ž. b. Ths s straghtforward. Ž b. Ž c.. Ths follows from Lemmas 4.2 and 4.3. Ž c. Ž a.. Suppose that Ž c. s satsfed. By Lemma 4.2 ŽŽ b. Ž a.., every unform submodule of M s essental n a drect summand of M. In partcular, by Lemma 4.1, every non-zero drect summand of M contans a non-zero unform drect summand. It follows from Theorem 3.4 ŽŽ b. Ž c.. that every local drect summand of M s a drect summand. In order to prove that M s a CS-module, t s suffcent to show that every complement submodule of M s a drect summand. Consder any complement submodule A of M. By Lemma 4.1, A contans a non-zero unform drect summand, so by Zorn s Lemma there s a maxmal local drect summand C n A wth all C unform. By the above, C C s a K K drect summand of M, hence of A, so ACBfor some B. Clearly B s a complement submodule of M Žcf. 6, p. 6.. If B 0, by Lemma 4.1, B contans a non-zero unform drect summand C. Then Ž C. K C s agan a local drect summand n A, whch s a contradcton to the maxmalty of K C. Therefore B 0, and so A s a drect summand of M whch proves our clam. The fact that the decomposton M IM complements drect summands follows from Theorem 3.4. Our fnal applcaton concerns the class of quas-dscrete modules, whch may be regarded as a generalzaton of projectve modules over rght perfect rngs. Before statng the result, we frst recall some basc defntons. A submodule A of a module M s called small n M f A B M for any proper submodule B of M. A module H s called hollow f every proper submodule of H s small n H. Followng Mohamed and Muller 12, p. 57, a module M s defned to be quas-dscrete provded t satsfes the followng two propertes: Ž D. For every submodule A of M, there s a 1

18 466 NGUYEN VIET DUNG decomposton M M1 M2 such that M1 A and A M2 s small n M; Ž D. 3 If M1 and M2 are drect summands of M wth M1 M2 M, then M1 M2 s agan a drect summand of M. By a result, due to Oshro Žsee, e.g., 12, Theorem 4.15., every quas-dscrete module s a drect sum of hollow modules, and so a natural queston arses: When s a drect sum of hollow modules quas-dscrete? Mohamed and Muller 12, Theorem 4.48 provded an answer to ths queston: Let M IM be a drect sum of hollow modules. Then M s quas-dscrete f and only f the followng three condtons hold: Ž. a M s jm j-projectve for every I; Ž b. M M complements drect summands; Ž c. I Every local drect summand n M s a drect summand. However, they also commented n 12, pp. 74 and 55 that t s hard to verfy these condtons n concrete cases, and so n full generalty the queston stll remans open. We can see now, from our Theorem 3.4, that f a decomposton M IM complements drect summands then every local drect sum- mand of M s a drect summand. Therefore, Mohamed and Muller s result can be substantally mproved as follows. COROLLARY 4.5. Let M IM be a drect sum of hollow modules. Then M s a quas-dscrete module f and only f M s jm j-projecte for eery I and the decomposton M IM complements drect sum- mands. ACKNOWLEDGMENTS Ths paper was wrtten durng a stay of the author at the Unversty of Udne and the Unversty of Glasgow, supported by the Italan CNR and the Royal Socety of London, respectvely. The author thans these nsttutons for ther hosptalty and the fnancal support. Specal thans are due to hs hosts, Professor A. Facchn and Professor P. F. Smth, for ther nd help. Fnally, the author s very grateful to the referee for several helpful suggestons. REFERENCES 1. F. W. Anderson and K. R. Fuller, Modules wth decompostons that complement drect summands, J. Algebra 22 Ž 1972., F. W. Anderson and K. R. Fuller, Rngs and categores of modules, n Graduate Texts n Math., Vol. 13, Sprnger-Verlag, New YorHedelbergBerln, P. Crawley and B. Jonsson, Refnements for nfnte drect decompostons of algebrac systems, Pacfc J. Math. 14 Ž 1964., N. V. Dung, On ndecomposable decompostons of CS-modules, J. Austral. Math. Soc. Ser. A 61 Ž 1996., N. V. Dung, On ndecomposable decompostons of CS-modules, II, J. Pure Appl. Algebra 119 Ž 1997.,

19 INDECOMPOSABLE DECOMPOSITIONS N. V. Dung, D. V. Huynh, P. F. Smth, and R. Wsbauer, Extendng modules, n Ptman Research Notes n Mathematcs Seres, Vol. 313, Longman, Harlow, K. R. Fuller, On generalzed unseral rngs and decompostons that complement drect summands, Math. Ann. 200 Ž 1973., K. R. Fuller, On rngs whose left modules are drect sums of fntely generated modules, Proc. Amer. Math. Soc. 54 Ž 1976., K. R. Fuller and I. Reten, Note on rngs of fnte representaton type and decompostons of modules, Proc. Amer. Math. Soc. 50 Ž 1975., M. Harada, Factor categores wth applcatons to drect decomposton of modules, n Lecture Notes n Pure and Appled Mathematcs, Vol. 88, Deer, New Yor, I. Herzog, A test for fnte representaton type, J. Pure Appl. Algebra 95 Ž 1994., S. H. Mohamed and B. J. Muller, Contnuous and dscrete modules, n London Math. Soc. Lecture Note Seres, Vol. 147, Cambrdge Unv. Press, Cambrdge, B. L. Osofsy, A generalzaton of quas-frobenus rngs, J. Algebra 4 Ž 1966., H. Tachawa, QF-3 rngs and categores of projectve modules, J. Algebra 28 Ž 1974., R. B. Warfeld, Jr., A KrullSchmdt theorem for nfnte sums of modules, Proc. Amer. Math. Soc. 22 Ž 1969., B. Zmmermann-Husgen and W. Zmmermann, Classes of modules wth the exchange property, J. Algebra 88 Ž 1984., B. Zmmermann-Husgen and W. Zmmermann, On the sparsty of representatons of rngs of pure global dmenson zero, Trans. Amer. Math. Soc. 320 Ž 1990.,