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The Torsion Submodule of a Cyclic Module Splits Off

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The Torsion Submodule of a Cyclic Module Splits Off Can. J. Math., Vol. XXIV, No. 3, 1972, pp. 450-464 THE TORSION SUBMODULE OF A CYCLIC MODULE SPLITS OFF MARK L. TEPLY A prominent question in the study of modules over an integral domain has been: "When is the torsion submodule t(A) of a module A a direct summand of A ?" A module is said to split when its torsion module is a direct summand. Clearly, every cyclic module over an integral domain splits. Interesting splitting problems have been explored by Kaplansky [14; 15], Rotman [20], Chase [4], and others. Recently, many concepts of torsion have been proposed for modules over arbitrary associative rings with identity. Two of the most important of these concepts are Goldie's torsion theory (see [1; 12; 22]) and the simple torsion theory (see [5; 6; 8; 9; 23], and their references). Both Goldie's torsion theory {^,JV) and the simple torsion theory (j^, J^~) are hereditary "torsion theories" in the sense of Dickson [5] and have an associated topologizing and idempotent filter of left ideals in the sense of Gabriel [10] (also see [16]). The main purpose of this paper is to continue the study of the splitting properties of these two types of torsion for modules over a commutative ring. The first two sections are devoted to terminology and preliminary results, which relate splitting properties to the existence of idempotent left ideals. It is interesting to note that the "idempotence" techniques, which are devel­ oped in § 2, work for both the Goldie torsion theory and the simple torsion theory, but not for hereditary torsion theories in general. The major result of § 3 (Theorem 3.3) proves the equivalence of the following three properties for a commutative ring R such that R £ ^\ (1) ^ (Rx) is a summand of each cyclic module Rx; (2) every closed ideal / of R has the form / = Re 0 S, where e2 = e and where 5 is a direct sum of simple R- modules or zero; (3) & (M) is a summand of each module M such that M/& (M) is a direct sum of cyclic modules. (It was shown in [23, p. 272] that there is no loss of generality in assuming R Ç ^Y,) This result generalizes the fact that every cyclic module over an intergral domain splits. In § 4 it is proved (Lemma 4.1) that if Sf (Rx) is a summand of each cyclic module Rx and R G J^~, then & = S^ for R^Jé. This enables us to see (Theorem 4.3) that (j^7, J^~) splits cyclic modules over a commutative ring R if and only if R is a direct sum of a semi-artinian ring and finitely many integral domains Di such that S^ = 5^ for DicS$. As corollaries we are able to obtain several known results on splitting for (5^,^"). Moreover, we are able to characterize the Received May 11, 1971. 450 Downloaded from https://www.cambridge.org/core. 27 Sep 2021 at 11:18:59, subject to the Cambridge Core terms of use. TORSION SUBMODULE 451 commutative rings R for which either of the following properties hold: (1) y {M) is a summand of each finitely generated i^-module M (Theorem 4.6); (2) (j^, J^~) is stable and has the bounded splitting property (see definitions below and Corollary 4.5). 1. Definitions and preliminary results. In this paper all rings R are associative with identity. Unless stated otherwise, all modules will be unitary left i^-modules, and the category of left i^-modules will be denoted by R*Jé. A torsion theory (£T,&~) is a pair of subclasses («^",^") of R^é satisfying the following conditions: iX)Tr\ JF = o. (2) B C A and A G T imply A/B G T. (3) B Ç A and A £ & imply B £ ^. (4) For each A Ç /j^, there exists an exact sequence 0-±T->A-> F-+Q such that T £ ^ and F G #". Then jF is called the torsion class, and J^ is called the torsionfree class. 3T is closed under homomorphic images, direct sums, and the extensions of one module in^7" by another.^ is closed under submodules, direct products, and extensions of one module in J^~ by another. Each module A £_R^ has a (necessarily unique) largest submodule in^T~; this submodule is denoted by f (A ). A class is called hereditary if it is closed under submodules; and a torsion theory (c^J^) is called hereditary if &~ is a hereditary class. A hereditary torsion theory (&~,&~) is uniquely associated with a topologizing and idem- potent filter of left ideals F{ST) = {I\R/I Ç^}. We shall use F(<T) as our standard notation for the filter associated with J?7". For further properties of (<^"~, i^~) the reader is referred to [5; 13; 16; 23]. Properties of filters may be found in [9; 10; 13; 22]. In this paper, we deal with two important hereditary torsion theories, namely Goldie's torsion theory (&,JV) and Dickson's simple theory (j^7, J^~). Goldie's torsion class & for R<Jt (see [1 ; 12; 22]) is the smallest torsion class containing all modules B/A, where A is an essential submodule of B. In case R is an integral domain, then & coincides with the usual torsion class. F(&) is the smallest filter containing all the essential left ideals of R. The torsionfree class y¥ associated with & is precisely the class of non-singular modules. In case R has zero left singular submodule, then & coincides with the E(R)- torsion class defined by Jans [13, p. 1255]. Dickson's simple torsion classa for R<J( (see [5;6;9]) is the smallest torsion class containing all the simple modules. If R is a Dedekind domain, then Sf coincides with the usual torsion class for R^é. For a commutative Noetherian ring, 5f also coincides with Matlis' class of modules with "maximal orders" (17). S^ has been useful in the study of Lowey series (see [8]). F(S^) is the Downloaded from https://www.cambridge.org/core. 27 Sep 2021 at 11:18:59, subject to the Cambridge Core terms of use. 452 MARK L. TEPLY smallest filter containing the maximal left ideals of R, and the torsionfree class J^~ associated with S^ is precisely the class of modules with zero socle. Let (<^~' ,^) be a hereditary torsion theory. A module A is said to split (relative to (IT^)) \i^"(A) is a summand of A. We say that (!T ,&") has the cyclic splitting property (CSP) if every cyclic module splits. (^, J^~) is said to have the finitely generated splitting property (FGSP) if every finitely generated module splits. We say that (^,&~) has the splitting property (SP) if every module splits. A module A is said to have bounded order if I A = 0 for some / G F(&~). Then (ST ,£F) is said to have the bounded splitting property (BSP) if every module A, whose torsion submodule $~ (A) has bounded order, splits. If {<3yJ/) has CSP (respectively, FGSP,SP) iovRJt, then R = & (R) 0P', R' G ^K, and (^,^) for ^^f has CSP (FGSP, SP). A similar result holds for (y,#~). So in characterizing the rings R for which (^,«yK) has CSP r (FGSP, SP) for R*Ji, it is sufficient to study R . Hence there is no loss of gener­ ality in assuming R G <JV . Similarly, in studying splitting for (^ ,^) we may assume R G ^ without loss of generality. Moreover, if R is a commutative ring such that R = © £"=i P„ then (&,J^) for ^ has CSP (FGSP, BSP, SP) if and only if, for each i = 1, 2, . , n, (^,JV) for RlJt has CSP (PG5P, 5SP,SP). We now record an elementary, but basic, result. PROPOSITION 1.1 Let (37~',^) be a hereditary torsion theory for R*Jé. Then the following statements are equivalent: (1) (F,^) has CSP. (2) ^~(M) is a summand of each module M such that M/3T(M) is a direct sum of cyclic modules. Proof. (2) => (1) is obvious. (1) =» (2). Let M/3T(M) be a direct sum of cyclic modules. Since Ext* (© EaCA, T) ^ IIa^ ExtB(Rxa, T) for every T £ $~, it is sufficient to show that Ext^Px, T) = 0 for all Rx £ J^~ and P G ^\ Consider the exact sequence (E) 0^r4 Y^Rx-^0. Then x(P) =^"(^). Choosey £ F such that a(y) = x. From (1) we see that Ry = $~(Ry) 0 P, and hence Y = X(P) + P. Since X(P) C\F ^^ C\ #"'= 0, the last sum is direct and (E) splits. Therefore, RxtR(Rx, T) = 0 as desired. Finally, we will make frequent use of the following well-known result (see [11, p. 58] for a proof): LEMMA 1.2. Let Rbe a commutative ring. If I is a finitely generated idempotent ideal of R, then I = Re, where e2 = e. Downloaded from https://www.cambridge.org/core. 27 Sep 2021 at 11:18:59, subject to the Cambridge Core terms of use. TORSION SUBMODULE 453 2. Lemmas on splitting and idempotence. Let (J~,<^) be a hereditary torsion theory with the cyclic splitting property (CSP), i.e. ^(Rx) is a summand of each cyclic module Rx. This condition places restrictions on left ideals / such that R/I GcF. In case (&~ ,&~) is the simple torsion theory (y ,^), we show that such left ideals / must be idempotent. And in the case where (ST ,^) is Goldie's torsion theory (^ ,J/), we show that CSP and R/I ^^V imply that I = P @A, where A is semi-simple or zero. If R is a commutative ring, then ,4=0, i.e. / is idempotent.
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