VietnamJournal of Mathematics 25:l (1997) 65-73 VIe trnannrjolrrrrmall of MI,A.1I]Ht lE M A]f ltCS q Springer-Verlag 1997

ExtendingProperty for Finitely GeneratedSubmodules

Le Van Thuyet (Jniuersity of Hue, Hue, Vietnam

R. Wisbauer (Iniuersityof Diisseldarf,D,iisseldorf, Germany

ReceivedJanuary 25, 1996

Abstract.A modlle is called extendingif everysubmodule is essentialin a direct sum- mand.More generally,the extendingproperty can be restrictedto certainclasses of sub- modules,e.g. uniform submodules, semisimple submodules. In thispaper, we considerthe extendingproperty for the classof (essentially)finitely generated submodules. We study propertiesof thistype of modules,decompositions and the relationship to finitelypresented modules.

1. Preliminaries

Throughout this paper,Rwill denotean associativering with unit and R-Mod the categoryof unital left R-modules.Morphism of left modulesare written on the right. For an R- M, olMl denotesthe full subcategoryof R-Mod whose objectsare submodulesof M-generatedmodules. A module M is said to havefinite unifurm dimensionif M doesnot contain an infinite direct sum of nonzerosubmodules. A submoduleK of M is calledessential in M if KnL # 0 for everynotzero submoduleL of M.In this case,M is called an essentialextension of K. A submoduleC of M is closedin M if and only if C is the only essentialextension of C in M. A module M is calledextending provided every closedsubmodule of M is a direct summand of M, or equivalently,every submoduleof M is essentialin a direct summandof M. M is called uniform-extendingif everyuniform submodule is essentialin a direct summandof M. Now we introduce somenew notions which generalizethe conceptof extend- ing modules. Recall that a module which has a finitely generatedessential submodule is said to be essentiallyfinitely generated(essentially finite fot short). In particular, this includesall finitely generatedmodules' Le Van Thuyet and R. Wisbauer

Definition l. A module M is called ef-extending if euery closed essentially finite submodule is a direct summand of M.

Definition 2. A module M is called f-extending if eueryfinitely generated submodule of M is essential in a direct summand of M.

Trivially, every extending module is ef-extending (f-extending). Moreover, every module whose finitely generated submodules are direct summands is ef- extending. In particular, every over a von Neumann rezular R is ef-extending. Hence, the obvious implications extending >ef-extending - f-extending + uniform extending are not reversible. The converse implications hold for modules with finite uniform dimension since for such modules uniform-extending implies extending (see [5, 7.g]). A ring Ris left PF (pseudo-Frobenius)provided R is an injective cogenerator in R-Mod. Equivalently, R is a semiperfect left self-injective ring with essential . A module M is continuous 1f M is extending and direct-injectiue, i.e., every submodule of M that is isomorphic to a direct summand of M is itself a direct summand of M. In [2], a module M is caltedfinitely continuous (for short f-continuous) If M is f-extending and direct-injective.

2. Decompositions of ef-Extending Modules

We begin with some elementary observations.

Lemma 2.1. Any closed submodule of an ef-extending module is also an ef-extending module.

Proof. suppose N is a closed submodule of M and M is ef-extending. Let x be a closed essentially finite submodule in N. Since N is a closed submodule in M, so by [5, 1.10],x is a closed submodule in M. Since M is anef-extendingmodule, x is a direct summand of M andhence in N. t

Now we consider the properties of decompositions of an ef-extending module.

Lemma 2.2. An indecomposablemodule is f-extending (dextending, extending) if and only if it is unform. Proof. Let M be an indecomposable f-extending module. For every x e M, x * 0, Extending Property for Finitely Generated Submodules 67

Rx is essentialin a direct summandA of M. Thus.,4 : M andthen Rx is essential in M. Hence.M is uniform. The other implicationsare obvious. I

Recallthat a directsum of submodulesof M, N: @^ N1e M, is saidto be a local direct summandof M if e, & is a direct summandof M for every finite subsetA e r\.

Corollary 2.3. Let M be an ef-extendingR-module. Assume that (a) euerylocal direct summandof M is a direct summandor (b) Enda(M) doesnot contqinan infinite set of orthogonalidempotents. ThenM is u direct sum of unifurm modules. Proof.In case(a), by [8, Theorem2.l7f,and in case(b), by [5, 10.4],M is a direct sum of indecomposablemodules. By Lemmas 2.1 and 2.2, M is a direct sum of uniform modules (in particular, in case(b), M is a finite direct sum of uniform modules). I

A moduleL e olMlis calledM-singular lf L = N I K for someN e olMl and, K is an essentialsubmodule of N. As is well known, everymodule L e olMl con- tains a largestM- which we denoteby Z2a(N).If ZM(N) :0, N is called nonsingularin olMl or non-M-singular.Note that, if M is non-M- singular,then everysubmodule of M hasa uniquemaximal essentialextension (see [0] for more details). Proposition2.4. Let M be a modulesuch that euerysubmodule of M has a unique maximalessential extension. Then M is ef-extendingif and only if M isf-extending. Proof. Let M be an f-extendingmodule. Let Lbe a maximal essentialextension of a finitely generatedsubmodule K. Then K is essentialin a direct summandfI of M.By assumption,H c L. From this, Z is a directsummand of M. Hence,M is ef-extending. T

For any m e M, we denote

l(m): {r e Rlrm:0}. The following observationis fundamental.

Proposition2.5. Let M be'an dextending R-module.If R satisfiesACC on left idealsof theform l(m),m e M, then M containsa mqximal local direct summand N : O,..lL, with Ni unifurmfor eachi e I. Proof. SinceR satisfiesACC on left idealsof the form l(m),m e M,we can choose a nonzeroelement m e M such that l(m) is maximal in {/(x)10 * x e M}. By the ef-extendingproperty of M,there existsa direct summandK of M such that Rmis essentialin K. SupposeK is decomposable.Then there exist nonzerosubmodules Kr and Kz of l( suchthat K:K1 @Kz.So we write m:mr+fti2 for some rfi1e K1,mzeKz. lf m1- 0, thenm:mze K2,and, RmoKl:0 givingKt :0, a contradiction.Thus, mr *0.It is easyto seethat l(m) c.l(m1).Hence,by the Le Van Thuyet and R. l[/isbauer maximality of m, l(m) : l(mt). Similarly, mz # 0 and' l(m) : l(mz). Because mt * 0 and Rm is essentialin K, there exist 11, 12e R suchthat

0 * rtmt : r2m: rz(mt+ mz): rzmtI rzmz.

From this,rzm2- 0, and hencer2el(m2)\l(m), a contradiction.So K is inde- composable. By Lemma 2.1, K is ef-extendingand by Lemma 2.2, K is uniform. It follows that any direct summandof M containsa uniform direct summand. By Zorn's lemma,M containsa maximallocal directsummand N: @;.1Ni, where N; is a uniform submodule of M for eachi e I. I

Corollary 2.6. Let M be an ef-extendingR-module. If R satisfiesACC on left ideals of theform l(m), me M, and euerymaximal local direct summandis essentially finite, thenM is afinite directsum of unifurmmodules.Consequently, M is extending. Proof. By Proposition2.5, M containsa maximal local direct summandN: @,.. N,. By [5, 8.1(1)],N is closedin M. Hence,N is a direct summandof M, say M:N@Nt, for somesubmodule N' of M.If N'+ 0, then by the sameargument as in the proof of Proposition2.5, Nt : U @ U' for some submoduleU, U', with U uni- form. Then N O U is a local direct summand,contradicting the maximality of N. Then N' : 0 and M : @,rrN; is a directsum of uniform submodulesof M. By assumption,M contains an essentialfinitely generatedsubmodule Z. It follows that thereexists a finite subset,I of 1 such that V - Or Nr. SinceZ is essentialin M, M: e"rN;, provingour claim. I

We also have somekind of decompositionof an ef-extendingmodule.

Proposition2.7. Let M be an dextending R-modulewhich is projectiuein olMl. ThenM : ei.r Mi, whereeach Mi is essentiallyfinite. Proof. By Kaplansky'stheorem (e.g. [13, 8.10]),the moduleM is a direct sum of countablygenerated submodules. Thus, without lossof generality,we may assume that M is countably generated,i.e., there exists a countable set of elements rnl, r/12,m3,... in M suchthat M : of,_tRmi. By hypothesis,there exist submodules M1,N1 of M suchthat M: Mt@Nr and Rm1 is essentialin M1. Then mz : flltr I n2. Supposethat n2 # 0. By Lemma 2.1, N1 is again an ef-extendingmodule, hence, there existsa direct summandMz of N1 which containsRn2 as an essentialsubmodule; moreover, Rmt*Rm2cMr@Mz. Continuingin this mannerwe obtain a directsum M1 @Mz@..'of sub- Extending Property for Finitely Generated Submodules 69

modules in the modules M such that

Rmt + Rmz -1 ... * Rm1,= Mt @ Mz@ ". @ Mr,

for all k e N. It follows that M : @;.w 4. Moreover, by construction, each submodule M; is essentially finite.

Corollary 2.8. Let M be an ef-extending R-module which is projectiue in olml and non-M-singular. Then M is a sum of finitely generated modules. Proof. By Proposition 2-7, M is a direct sum of essentially finite ef-extending modules. Since a non-N-singular module N which is projective in o[N] and has an essentially finite submodule is finitely generated (see [5, 4.7]), hence the desired proof follows. I

A module M is caTledn-injectiue if for any Lt, Lz e M with L1 o L2 : 0, there existsubmodules M1, M2 of M such that M : Mr @ M2 Lnd Li e Mi (i:1,2). Now we have a characteization of an ef-extending module via a property that is close to the property of a z-.

Theorem 2.9. For an R-module M, thefollowing conditions are equiualent: - (a) For any L1, L2 c M with Lt ^Lz 0 and L1 essentially finite, there exist submodules Mr, Mz of M such that M : Mr @ Mz, Lr is essential in M1 and L2 c M2, (b) M is an ef-extending module and wheneuerM : Mt @ Mz with Mr essentially finite, then Ml is M2-injectiue. Proof. (a) +(b). First, we prove that M is an ef-extending module.Let Abe a closed essentially finite submodule in M and L2 a complement of ,4 in M; then, by assumption (a), there exist submodules M1, M2 of M such that A is essential in Mt, Lz c. M2 and M : Mr @ Mz.It follows that L2 : Mz and A : Mt. So ,4 is a direct summand of M. Hence, M is an ef-extending module. Let M : Mt @ Mz with Mr essentially finite and N a submodule of M such that N ) M1 :0. Then by assumption (a), there exist submodules M' and,M! of M such that M : Mt@ M'l

and N c. M', M1 = Mi.Hence, by the Modular Law,

M', : Mi o M : Mi o (M1 @ Mz) : Mt @ (Mt, a M2) and M : M' @ Mr @ (Mi n A) : M' @ (Mi n Uz) @ Mt. By [5, 7.5],Mr is Mz-injective. (b) = (a). Let L1 be essentiallyfinite with a finitely generatedessential sub- module B and L2 c M such that Lt a L2 :0. We take the complementC of ,Bin M. This is to say that L1 aC:0. Then by Zorn's lemma,there exists maximal submoduleD e M suchthat D o C : 0 and D = B.Moreover. B is essentialin D. Le Van Thuyet and R. Wisbauer

So D is a closedessentially finite submoduleof M. SinceM is ef-extending,D is a direct summand of M, say M:D@F.

Since11 is alsoessential in D, D A L2 :0. By (b), D is F-injective.Hence, by 15,7.sl,there exists a submoduleP' of M suchthat M : D@ P/ and L2 e Pt. So (a) follows. I

3. Finitely X-F-(ef)-ExtendingModules

For any R-module M, denoteby Add M (add M, resp.)the full subcategoryof o[M] whose objects are direct summandsof (finite) coproductsof copiesof M. Of course,Add R is just the class of all projective R-modulesand it does not contain (R)-singular modules.It is clear that if M is projective in olMl, or M has no M-singular submodule,then Add M containsno M-singular modules(see [3, Lemma1.10]). An R-module M is said to be direct-projectiueif, for every direct summandX of M,every epimorphismM - X splits.M isZ-direct-projectiueif any coproduct of copiesof M is direct-projective.Of course,if M is projectivein olMl, then it is X-direct-projective. A module N is called finitely presentedin M if it is finitely generatedand in everyexact sequence 0*K-L--+N--0 in ofMl, with Z finitely generated,K is also finitely generated. lf M : R, then the class of all finitely presentedmodules is the class of all modulesof the form R" f H whereH c Rn finitelygenerated in [13,2.5].

Definition. We call an R-moduleM (finitely)E-f-extendingif any (finile) coprod- uct of copiesof M isf-extending. M is called (finitely) E-ef-extendingf any (finite) coproductof copiesof M is ef-extending.

Recall the following resultsabout regular rings and left and right hereditary serialArtinian rings.

Proposition3.1. For a ring R, thefollowing conditionsare equiualent. (a) R rs uonNeumann regular and @) eueryfinitely presentedleft (and right) R-moduleis projectiue. Proof. See113,37.61. I

Proposition3.2. For a ring R, thefollowing conditionsare equiualent: (a) R rs a left (and right) hereditaryserial Artinian ring and (b) R k fight nonsingularand euerynonsingular left R-moduleis projectiue. Extending Property for Finitely Generated Submodules 7l

Proof. See[7, Theorem5.23]. I

Now we consider properties of finitely x-f+xtending which are related to Propositions3.1 and,3.2.

Lemma 3.3. Let M be ef-extending(f-extending) and K an essentiallyfinite (a finitely generated,resp. ) submoduleof M suchthat MIK is non-M-singular.Then K is a direct summandof M. Proof. SinceM is f-extending,there exists an essentialextension R of K which is a direct summandof M. sav

M:K@V. From this

MlK - (Rlx)A v. since M lK is non-M-singular,it follows that K: K and then K is a direct sum- mand of M. The proof for ef-extendingis similar. I corollary 3.4. If R is a left finitely Z-f-extendingring, theneuery finitely presented nonsingularleft R-moduleis projectiue.

Theorem3.5. Let M be a module.We considerthe following conditions: (a) M isfinitely dextending, (b) euerymodule M, (a) (b) (c), and (c) eueryfactor of M" by any closedessentially finite submoduteof M" is in Add M. Thenfor euerymodule M, (a) e (b) <+(c). If M isZ-directprojectioe, then (c) +(a). Proof. (a)e (b). Bv Lemma 3.1, any direct summandof an ef-extendingmodule is also ef-extending. (a) =+(c). Let H be a finitely generatedsubmodule of Mh andE anymaximal essentialextension of f1. ThenfI is a directsummand of Mn, sayMn : tt @ K.lt is easyto seethat M" lE - K is in add M . (c) + (a). Let M be X-direct projective.Assume (c). Let H be a finitely gen- erated submoduleof M" and E any maximal essentialextension of f1. Then bv (c), M" lE is in Add M. SinceM is X-directprojective, E is a direct summandoi M".It followsthat M" is ef-extending. For the f-extendingproperty, we obtain the following:

Theorem3.6, Let M be an R-module.We considerthe following conditions: (a) M is2-f-extending, (b) M isfinitely 2-f-extending, 72 Le Van Thuyet and R. Wisbauer

(c) euery factor of M" by some closed essentially finite submodule of M" is in Add M. (d) eueryfactor of M" by a finitely generatedsubmodule is a direct sum of a module in Add M and an M-singular module, (e) euery non-M-singular factor of M" by any finitely generated submodule is in Add M. Then we haue the following implications: (l) For euery module M, (a)<+ (b) = (c), (d) - (e). Q) If M is Z-direct projectiue, then (c) + (a), (c) - (d). Proof. (a) +(b) is trivial. (b) - (a). Let H be a finitely generated submodule of M6), where A any rndex set. Then there exists n e N such that H c M'. By (b), fI is essential in a direct summand,E of M".It is easy to see that,F is also a direct summand of lr(^). So (a) follows. The other implications are proved similarly to Theorem 3.5. I

Corollary 3.7. (l) For a ring R, the following conditions are equiualent: (a) R ls left finitely \-dextending; (b) eueryfinitely generatedprojectiue is left R-module ef-extending; (c) euery factor of R" by any closed essentially finite submodule of R" is projectiue. (2) For a ring R, the following conditions are equiualmt: (a') R is left2-f-extending; (b') R is left finitely 2-f-extending; (ct) euery factor of R" by some closed essentially finite submodule of R' is projectiue. (3) If RR is nonsingular, then all conditions in (l) and (2) are equivalent. (4) One of the aboue conditions in (L) and (2) - (d) + (e) are as follows: (d') eDeryfinitely presented left R-module is a direct sum of a projectiue module and a singular module; (e') euery nonsingularfinitely presented left R-module is projectiue.

Acknowledgements.This paperwas written during a stay of the first author at the Heinrich- Hei+ie University of Diisseldorf,Germany. He wishesto thank the DAAD Foundationfor financial supportand the membersbf the Departmentof Mathematicsfor their hospitality. The authors also wish to thank Dr. Brodskii and Dr. M. Salehfor many stimulating dis- cussionsduring the preparationof this paper.

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