Math. Ann. 184, 300~308 (1970)

On the Decomposition of Modules and Generalized Left Uniserial Rings

PHILLIP GRIFFITH

1. Introduction

Combining theorems of K6the [10] and Cohen and Kaplansky [2] one obtains

Theorem (K6the-Cohen-Kaplansky). A commutative ring R has the property that each of its R-modules is a direct sum of cyclic modules if and only if R is an artinian principal ideal ring.

Nakayama [12], on the other hand, showed that a larger class of rings (generalized uniserial rings) has the above property. Moreover, Nakayama [I3] demonstrated that even the notion of a generalized uniserial ring does not characterize the class of rings all of whose modules are direct sums of cyclic modules. More recently, Faith [3] further studied artinian principal ideal rings and showed that this property is equivalent to several other properties of rings. For example, Faith proved that the property of a ring being a left uniserial, quasi-Frobinius ring (= Q.F. ring) is equivalent to a ring being an artinian principal ideal ring.

In this paper we wish to obtain a theorem concerning the decomposition of modules into direct sums of cyclic modules which holds for noncommutative tings and also contains the results of the aforementioned authors. In our con- siderations, we restrict our attention to those rings R all of whose left modules are direct sums of cyclic modules having unique composition series. Theorem 4.1 characterizes the class of rings having the above property. This class includes the class of generalized uniserial rings. We show that a generalized left uniserial, Q.F. ring is equivalent to the ring being Q.F. modulo powers of its radical. Furthermore, these results can be interpreted in terms of the vanishing of a certain "pure left global dimension" (defined in Section 3) of a ring. Indeed our Theorems 4.1 and 4.3 bear a resemblance to the Wedder- burn-Artin theorem as presented in [8] where simple module is replaced by cyclic module with a unique composition series and global dimension is replaced by our pure global dimension. Theorem 4.3 also shows that a com- mutative ring for which each module is a direct sum of finitely generated modules is necessarily an artinian principal ideal ring. Finally, in Section 2 we obtain some information on commutative rings having the property that each of its modules is a direct sum of countably generated submodules.

Decomposition of Modules 301

In what follows, all rings are associative with identity and all modules are unital. An R-module M is called finitely presented provided there is a finitely generated free module F and an epimorphism F--~M whose kernel is finitely generated. After Kaplansky, a left R-module M will be called cyclically presented if M,,~ R/Ra for some a e R. The notation E(M) and P(M) is used to denote the injective envelope and projective cover (if the projective cover exists, see [1]), respectively. Also socM indicates the socle of M. If R is a ring then J denotes its Jacobson radical and R J# and R~// are used to denote the category of left R-modules and the class of cyclic left R-modules having unique composition series, respectively. After Faith [3] a ring R will be called a generalized left uniserial ring if R is left artinian and each principal indecomposable left ideal has a unique composition series as a left R-module. A left uniserial ring is a generalized left uniserial ring which is a direct product of finitely many primary rings. A Q.F. ring is characterized by the property that it is left artinian and left self injective (see [73).

2. Countably Generated Modules

A module which is a direct sum of countably generated modules will be referred to as a d.s.c. The author's Theorem 2.2 [5] may be rephrased to read

Theorem 2.1. Each fiat left R-module is a d.s.c, if and only if R is left perfect, that is, if and only if each fiat left R-module is projective 1

Actually the techniques of [5] may easily be extended to show that there is an infinite cardinal number d such that any flat left R-module is isomorphic to a direct sum of modules each of which can be generated by d elements if and only if R is left perfect. Along this same direction, Faith and Walker [4] proved that there is a cardinal number d such that each injective left R-module is contained in a direct sum of modules generated by d elements if and only if R is left noetherian. In case each left R-module is isomorphic to a direct sum of modules which can be generated by d elements, we can strengthen Faith and Walker's conclusion in one direction.

Theorem 2.2. I f d is a fixed cardinal number and if each left R-module is isomorphic to a direct sum of modules which can be generated by d elements then R is left artinian. In particular, if each left R-module is a d.s.c., then R is left artinian.

Proof. In view of the above discussion R is both left perfect and left noetherian which shows from [1] that R is left artinian.

Before leaving this section we wish to show that a particular class of com- mutative local artinian rings each has a nonfinitely, countably generated indecomposable module. This result will be applied in Section 4. Let R be a commutative local artinian ring with Jacobson radical J = R a G R b where Ra and Rb are both simple R modules; hence j2 = 0. If k denotes the field R/J

1 For information on left perfect rings, see Bass [1].

302 Ph, Griffith:

and if M is an R-module, then the R-modules M / J M and J M are naturally vector spaces over k and the mappings f , , J b : M / J M - , , J M are k-linear transformations where f , (x + J M) = ax and f b(x + J M) = b x. It is clear that M is an indecomposable R-module if there are no non-trivial decompositions M / J M = V1GV 2 and J M = IJ/I(gW 2 such that f,(VOC__W1, fb(P~)c= W 1 and fa(V2) c__ W2, fb(V2) _c_ W 2. Let F = @ Re, denote a free R-module, let K be the

n < ¢ 0

submodule generated by the set [ben+ 1 -aeJ,<,o and let M = F/K. Clearly M is nonfinitely, countably generated, J M = J F / K and M / J M = F / J F . Let x . = e , + J F e F / J F and y . = b e n + K e J F / K = J M for n = l , 2 . . . . . If V= M / J M and W = JM, we observe that [xn],<,o and [YJ.<,o are bases for the k-vector spaces V and W, respectively, and moreover the linear trans- formations f , and fb (defined above) have the property that fa(x.) = y~+ 1 and fb(x,) = yn for each n. Suppose that V = 1/1 (9 1/2 and W = W1 (9 W2 where f~(Vi)C= VV~ and fb(V~)~ W~ for i = 1, 2 where V 1 + 0 + V2. It follows that f~(V1)= W 1 or f , (V2)= W2 since dimk(W/Imf~)= 1. We may suppose that f~(V1) = 1411. Define t/(V0 to be the smallest positive integer n such that a vector v e I/1 has a nonzero coefficient at x~. Define t/(W~) similarly with respect to the basis [Y~m<~,- Since f~(V0=W1, it follows that t / ( ~ ) = t ] ( V ~ ) + l . But fb(VOC= W~ implies that t/(W~)=t/(V0 which contradicts the preceding state- ment. It is now apparent that M is indecomposable 2. With the above facts established, we now prove

Theorem 2.3. I f R is a commutative artinian ring which is not a principal ideal ring, then R has a nonfinitely, countably generated indecomposable R-module.

Proof. Since R is necessarily a finite direct sum of local rings, we may suppose that R itself is a local ring. Moreover, since R is not a principal ideal ring, then neither is R/J 2. It follows that R/J 2 and hence R has a homomorphic image R o which satisfies the description of the preceding paragraph. Hence R0 has a nonfinitely, countably generated indecomposable module M and thus so does R.

We unfortunately leave open the complete characterization of those commutative rings R for which each R-module is a d.s.c.

3. Purity, Pure Global Dimensions and a Decomposition Theorem

Using the terminology of Warfield [14], we say that a submodule A of a left R-module B is a pure (r.d.-pure) 3 submodule, if for any finitely presented (cyclically presented) module F, the natural homomorphism Hom(F,B) ~ H o m ( F , B/A) is surjective. An equivalent condition is that for any finitely presented (cyclically presented) right R-module M, the natural homomorphism M ® A - - , M ® B is injective. It is also equivalent to say that A is r.d.-pure in B

2 The above technique is similar to one used in [6]. 3 r.d. =relatively divisible.

Decomposition of Modules 303

if and only if rBc~A = rA for each r E R. A module P is called pure projective (r.d.-projective) if for any module A and pure (r.d.-pure) submodule B, any homomorphism of P into A/B can be lifted to a homomorphism of P into A. Warfield [14] shows that any ring has enough pure projectives (r.d.-projectives) and that a module P is pure projective (r.d.-projective) if and only if it is a direct summand of a direct sum of finitely (cyclically) presented modules. In what follows we shall confine ourselves to purity. However, the situation is entirely analogous for r.d.-purity. It is easy to verify that the pure short exact sequences form a proper class in the sense of [1 t], from which it follows that one can define the corresponding subfunctors of Ext,(A, B) (which we denote by Pext"(A, B) and PexL"d(A, B), respectively) in such a way that one obtains the usual long exact sequences [11, pp. 371--375]. Hence we define the pure-projective dimension PpD(A) of an R-module A to be the smallest positive integer n such that Pext" +1 (A, C) = 0 for all C. If no such n exists, then PpD(A) = oQ. Since by Proposition 1 [14], there are enough pure projectives, it is easily shown that PeD(A) may be obtained in the usual manner from pure-projective resolutions of A. Finally, we define the pure-left global dimen- sion 1. GeD(R) of a ring R to be sup {PeD(A) [ A is a left R-module}. The notation corresponding to r.d.-purity is PrdD(A) for A a left R-module and 1.GrdD(R ) for the r.d,-left global dimension of R.

Before proving our main results of this section, we establish two elementary facts concerning purity and r.d.-purity.

Lemma 3.1. Let R be any ring and let A be a left R-module. (a) I f B is a submodule of A and C a pure (r.d.-pure) submodule of A such

that C ~ B c_ A and such that B/C is pure (r.d.-pure) in A/C, then B is a pure (r.d.-pure) submodule of A.

(b) I f {B~}z~A is an ascending chain of pure (r.d.-pure) submodules of A, then B = &~A B~ is a pure (r.d.-pure) submodute of A.

Proof. (a) Let F be a finitely (cyclically) presented left R-module. By hypoth- esis the natural homomorphisms Horn(F, A)-, Horn(F, A/C) and Hom(F, A/C) --* Horn(F, A/B) are both epimorphisms. Hence the commutative diagram

Horn (F, A) -Horn (F, A/C)

Hom (F, A) ~ ,Hom ( f , A/B)

yields that v is an epimorphism. Thus B is a pure (r.d.-pure) submodule of A. Part (b) follows easily from our previous discussion of purity and r.d.-purity and the fact that direct limits commute with tensor products.

Theorem 3.2. Let R be any ring and let ~ be a class of finitely presented left R-modules. I f each nonzero left R-module contains a copy of a nonzero module in ~ as a pure submodule, then each left R-module is a direct sum of copies of modules in ~ .

304 Ph. Griffith:

Proof. Let M be any nonzero left R-module and let [F3.]).e A be an independent family of nonzero submodules which are isomorphic to modules in o~ and which are also maximal with respect to the property that F = @ Fa

2~A

is a pure submodule of M. Our hypothesis implies that A is nonvoid and Lemma 3.1 (b) guarantees that there is a collection maximal with respect to the above property. Suppose that F 4 = M. Then by hypothesis there is a sub- module A of M such that A properly contains F, A/F is pure in M/F and A/F is isomorphic to a nonzero module in o~. Since each module in ~ is pure projective, it follows that A = F O B where B is isomorphic to a nonzero module in ~ . By Lemma 2.1 (a), A is a pure submodule of M and hence the collection [B]v[Fz]z~ A is an independent family of nonzero submodules from ~- such that their direct sum is a pure submodule of M. This contradicts the maximality of the family [Fa]ge A. Thus F = M.

Theorem 3.3. Let R be a left artinian rin9 and let Y be a class of cyclic left R-modules containing the simple left R-modules. I f ~ is closed under the opera- tions of takin 9 submoduIes, homomorphic images, injective envelopes and pro- jective covers, then each left R-module is a direct sum of modules in .~.

Proof. Let M 4 = 0 be in RJtL. Our hypothesis guarantees that M contains a nonzero submodule A from .,~. Indeed, we may choose A to have maximum length among the modules in o ~ isomorphic to submodules of M. The mono- morphisms i: A ,-~M and zc :A ~--~E(A), where i denotes inclusion and z~ is the injection of A into its injective envelope, induce the commutat ive diagram

A?- i ) M

.I / E(A)

Note that H = I m f 3__ Im rc .~ A and observe that A is a direct summand of M if H = I m re. Since A is in o ~ , we have by hypothesis that E(A) is in ~ from which it follows that H and P(H) are also in Jr. Therefore, there is a commutative diagram

where C = I m ~o has the property that f ( C ) = H and that C is in i f . Hence length (C) > length (H) > length (A) if H,4= l m re. But this contradicts our choice of A. Therefore A is a direct summand of M and thus by Theorem 3.2 each left R-module is a direct sum of modules from o~. In the next section we shall apply this theorem.

D e c o m p o s i t i o n o f M o d u l e s 305

4. A Generalization of a Theorem of KiRhe, Cohen, and Kaplansky

Our first theorem of this section characterizes those rings R for which 1. G,d(R ) = 0. Recall that a module C in RJ/4" is a cogenerator in case each module M in R~' can be mapped monomorphically into a direct product of copies of C.

Theorem 4.1. Let R be any ring. Then the following are equivalent: (1) R is left artinian and Roll is closed under the operations of taking injective

envelopes and projective covers. (2) t. GraD(R) = 0 and each indecomposable R/Ra is in Rag. (3) Each left R-module is a direct sum of modules in Rag. (4) R is a generalized left uniserial ring and there are C1,.. . , C, in Roll such

that C = 0 C i is an injective cogenerator of RJt. i = 1

Proof. (1)o(3) follows from Theorem 3.3 and the fact that Rag is closed under the operations of taking submodules and homomorphic images.

(3)~(4). Each principal indecomposable left ideal of R is necessarily iso- morphic to a module in Rag. Since R is also left artinian by Theorem 2.2, it follows that R is a generalized left uniserial ring. Let $1, ..., S, be a complete set of representatives of the isomorphism classes of simple left R-modules and set Cg = E(Si) for i = 1, ..., n. By hypothesis Ci is in Rq/for each i since E(Si) is necessarily indecomposable for each i. It follows by Proposition 1 E4] that

C = (f~) Ci is an injective cogenerator of R ~ . i = 1

(4)~ (1). We have that R is left artinian from the definition of a generalized left uniserial ring. Let A be in gag. Then socA is simple and hence E(A) is an indecomposable direct summand of C. Since the modules in Rag have local endomorphism rings, it follows by Corollary 8.4 [15] that E(A) is in Rag. We also have that A/JA ~ P(A)/JP(A) is a simple module since A is in gag. It is therefore easily seen that P(A),~ Re for some primitive idempotent e of R. Since R is a generalized left uniserial ring, we have that P(A) is in Rag. Thus Rag is closed under the operations of taking injective envelopes and projective covers. Thus (1), (3) and (4) are equivalent.

(3)~(2). Let A be in Rag" The above argument shows that P ( A ) ~ R e for some primitive idempotent e of R. Since Jie/J~+le is necessarily simple or zero for each i, it follows that each submodule of Re is cyclic. Hence A ~ Re/Rre for some r e R. This implies that A ,~ R /Rx where x = re + (1 - e), that is, A is cyclically presented. It follows that each left R-module is a direct sum of cyclically presented modules which implies by Corollary 1 [14] that each left R-module is r.d.-projective. Thus 1.G, aD(R)=0.

(2)=>(3). Since 1.GraO(R)=O and since, as mentioned above, there are enough r.d.-projectives, it follows by Corollary 1 [14] that each left R-module is a direct summand of a direct sum of cyclically presented left R-modules. By Theorem 2.2 and a theorem of Kaplansky [9], we have that R is necessarily

21 Math. Ann. 184

306 Ph. Griffith:

left artinian. We now have that each left R-module is a direct summand of a direct sum of modules of the form Re/Rre where e = e 2 is primitive and, in addition, that each such Re/Rre has a local endomorphism ring. From Corol- lary 8.4 [15], we obtain that each left R-module is a direct sum of modules of the form Re/Rre where e = e 2 is primitive. Hence each indecomposable module is the homomorphic image of a principal left ideal that is generated by a primitive idempotent. This property is necessarily inherited by R/J 2. It follows from Nakayama's proof of Theorem 3 [13] that Re/J2e is in R°~ ' for each primitive idempotent e. Moreover, Nakayama's proof of Lemma 3 [13] now shows that Re is in R~// for each primitive e = e z. Thus from the above we have that each left R-module is a direct sum of modules in Rg/. This completes the proof.

Since Nakayama [12] showed that each left module over a generalized uniserial ring R is isomorphic to a direct sum of modules in R~//, it follows that the class of rings satisfying Theorem 4.1 contains the class of generalized uniserial rings. We do not know if this containment is proper. The next theorem is a specialization of Theorem 4.i and contains our characterization of generalized left uniserial, Q.F. rings. A left module M is called torsionless provided M can be embedded (as a left R-module) into a direct product of copies of R. In the following R~* denotes the class of torsionless modules in Rq/.

Theorem 4.2. Let R be any ring. Then the following ape equivalent: (1) R is left artinian and R~* is closed under the operations of takin9 injective

envelopes and projective covers. (2) I. GrdD(R) = 0 and each indecomposable cyclic left R-module is in Rq/*. (3) Each left R-module is a direct sum of modules in R~?I *. (4) R is a 9eneralized left uniseriat, Q.F. ring. (5) R/J k is Q.F. for each k. (6) The left-right symmetry of (I)---(4).

Pro@ The implications (I)~(3), (4)~(1) and (2),~=>(3) are clear from the proof of Theorem 4.1. Thus the equivalence of (1)--(4) will be established when we show that (3)~(4). Therefore assume that each left R-module is a direct sum of modules in Rd//* and let E be an indecomposable injective left R-module. Hence E is in Rqd* which implies that E is a direct summand of R since E is injective and also an essential extension of a simple module. Therefore E is also projective. Since each injective left R-module is a direct sum of indecom- posable injective modules (recall that R is left artinian by Theorem 2.2), it follows that each injective left R-module is projective. By Theorem 5.3 [4], R is a Q.F. ring. Since it is clear that R is a generalized left uniserial ring, we have completed the equivalence of (t)--(4).

Since (5)~(4) using that R/J 2 is Q.F., it suffices to prove (4)~(5) in order to complete the proof of the theorem.

(4)~(5). Assume that R is a generalized left uniserial, Q.F. ring. Since (3) and (4) are equivalent, we have that each left R-module is a direct sum of

Decomposition of Modules 307

modules in R ~ . Clearly the ring/~ = R/J 2 inherits this same property, that is, each left R-module is a direct stma of modules in n~. Hence, for each primitive idempotent e, we have that length (Re/J2e) = 1 or 2 both as an R and R-module. Let E denote the R-injective envelope of Re/j2e. Since E is necessarily in- decomposable, ff~,,~Re'/le' where J2e'C__le' and where e' is a primitive idempotent of R. Hence length (E)= 1 or 2 both as an R and R-module. If length (Re/j2 e)= 2, then Re/J2e~E, that is, Re/J2e is ,~-injective. Suppose now that length (Re/j2e)= 1. This implies that Je = 0 and that Re is a simple projective R-module. If length (E)=2, then necessarily Re~Je ' /J2e ' as an R-module. Since Re is projective, we have that J2e' is a direct summand of Je' and hence that J2e'= O. Therefore Re ~ socRe' =Je' which implies that socRe' is a direct summand of Re' since R is a Q.F. ring. Hence Je'= 0 which implies that length (E)= length(Re'lie')= 1. But this is-a contradiction. It follows that length (E)= 1 and that Re/j2e,.~ F as R-modules. Hence Re/j2e is necessarily ,~-injective for each primitive idempotent e. Thus R is also a Q.F. ring and hence R is generalized uniserial. Since R is a Q.F. generalized uniserial ring, it follows that Re/f ie is R/J k injective; indeed

Re~J% = {x e E(Re/Jk e) : f i x = 0}

for e a primitive idempotent. Thus R/J k is Q.F. for each k. Following from the above theorem is our result for commutative rings

which is essentially the theorem of K6the, Cohen, and Kaplansky mentioned in the introduction. However, two additional facts of interest are obtained. These are that the pure-global dimension of a commutative ring and the r.d.- dimension must vanish simultaneously and that if each R-module (R commu- tative) is a direct sum of finitely generated modules then R must be an artinian principal ideal ring.

Theorem 4.3. Let R be a commutative ring. The the following are equivalent: (1) G, aD(R)=O. (2) Each R-moduleis isomorphic to a direct sum of cyclic modules. (3) G,,D(R)=O. (4) Each R-module is isomorphic to a direct sum of finitely generated modules. (5) R is an artinian principal ideal ring. (6) R is a generalized uniserial, Q.F. ring. (7) Each R-module is isomorphic to a direct sum of copies of ideals of R. Prooj. Clearly Theorem 4.1 and Theorem 4.2 show that (1)=>(2), (5)=~(6)

and (6)~(1). The Theorem of KBthe, Cohen, and Kaplansky gives (2)=>(5). Hence (1), (2), (5) and (6) are equivalent. Theorem 4.2 shows that (6)=>(7) since each module in Rq/* must necessarily be an ideal of R. To show that (7)=>(6) assume that each R-module is isomorphic to a direct sum of copies of ideals of R. By Theorem 2.2 R is artinian. It is also clear that each injective R-module is projective from which it follows from [4] that R is a Q.F. ring. Since R is necessarily a direct sum of finitely many local rings, we may also assume that R is a local ring. Since each R-module is necessarily torsionless, it follows that soc(R/d ~) is simple for each i. Thus R is a generalized uniserial ring. We now

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308 Ph. Griffith: Decomposition of Modules

have that (1), (2), (5), (6), and (7) are equivalent. Furthermore, it is clear that (2)=~(4)=~(3). Hence the proof is complete when we show that (3)=~(5). We assume that R is a commutative ring such that G p D ( R ) = O. It follows that each R-module is a direct summand of a direct sum of finitely generated modules. Hence, by Kaplansky's theorem [9] and Theorem 2.2, we have that R is artinian. It now follows that each R-module is a direct summand of a direct sum of indecomposable finitely generated modules. Since indecomposable finitely generated R-modules over artinian rings have local endomorphism rings, another application of Corollary 8.4 [15] shows that each R-module is in fact a direct sum of finitely generated modules. It follows from Theorem 2.3 that R is a principal ideal ring. The proof is complete.

References

1. Bass, H. : Finitistic dimension and a homological generalization of semi-primary rings. Trans. Amer. Math. Soc. 95, 466--488 (1960),

2. Cohen, I. S., Kaplansky, I.: Rings for which every module is a direct sum of cyclic modules. Math. Z. 54, 97--101 (195t).

3. Faith, C.: On K6the rings. Math. Ann. 164, 207- -2 t2 (1966). 4. - - Walker, E. A.: Direct sum representations of injective modules. J. of Algebra. 5. Griffith, P.: A note on a theorem of Hill. Pacific J. Math. to appear. 6. Hetter, A, Reiner, I.: Indecomposable representations. Itl. J. Math. 5, 314--323 (196t). 7. Ikeda, M. : A characterization of quasi-Frobenius rings. Osaka Math. J. 4, 203--210 (1952). 8. Jans, J. P.: Rings and homology. New York: Holt, Rinehart and Winston, i964. 9. Kaptansky, I.: Projective modules. Ann. of Math. 68, 371--377 (t958).

10. K6the, G.: Veraltgemeinerte Abelsche Gruppen mit hyperkomplexen Operatorenringen. Math. Z. 39, 31--44 ( t9M).

t l. Mac Lane, S. : Homology. Berl in-G6tt ingen-Heidelberg: Springer 1963. 12. Nakayama, T. : On Frobennisean algebras II. Ann. of Math. 42, 1--21 (1941). I3. - - Note on uni-serial and generalized uniserial rings. Proc. Imp. Acad. Tokyo 16, 285--289

(1940). i4. Warfield, R. B. : Purity and algebraic compactness for modules. To appear. 15. - - Decompositions of injective modules. To appear.

Prof. Ph. A. Griffith The University of Chicago Department of Mathematics Chicago, Illinois 60637, USA

(Received February 13, 1969)