Two Generalizations of Lifting Modules

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1 International Journal of Algebra, Vol. 3, 2009, no. 13, Two Generalizations of Lifting Modules Nil Orhan Ertaş Süleyman Demirel University, Department of Mathematics Çünür Isparta, Turkey Rachid Tribak Département de mathématiques Faculté des sciences de Tétouan B.P Tétouan, Morocco Abstract In this paper we introduce the notions of G 1 L-modules and G 2 L- modules which are two proper generalizations of lifting modules. Then we give some characterizations and properties of these modules. We also show that these two classes of modules are not closed under direct sums. Some special cases of direct sums of these modules are studied. Moreover, Harada rings are characterized by using G 1 L-modules. Mathematics Subject Classification (2000): 13C05; 16D50; 16D80 Key words: lifting modules, cosingular modules, small modules 1 Introduction Throughout this paper all rings have identity and all modules are unital right modules. Let R be a ring and M an R-module. A submodule S of M is called a small submodule (notation S M) if M S + T for any proper submodule T of M. In [5], Leonard defined a module M to be small if it is a small submodule of some R-module. He showed that a module M is small if and only if M is small in its injective hull. Let N be a submodule of M. The module M is called a lifting module if for every submodule A of M there exists a direct summand B of M such that B A and A/B M/B. In [10],

2 600 N. O. Ertaş and R. Tribak Talebi and Vanaja defined Z(M) = {U M M/U is small}. If Z(M) = 0 ( Z(M) = M), then M is called a cosingular (non-cosingular) module. Let X be a class of modules. Adopting the notation of [1], we denote by d*x the class of modules M such that each submodule N of M contains a direct summand K of M such that the factor module N/K belongs to X. Some general properties of the class d*x are given in [1]. Moreover, some properties of d*x have been studied in various special cases (see e.g. [1] and [8]). Our purpose is to study some properties of d*x for two special classes X of modules. In the first section we prove that if X is a class of modules closed under submodules, then the direct sum of two projective modules in d*x is also in d*x. In Section 3, we consider X 1 the class of small modules. For X 1, we associate the class d*x 1 with the property G 1 L. A module M is a G 1 L-module, if for every submodule N of M, there exists a direct summand K of M such that K is contained in N and the factor N/K is a small module. It is shown that the calss of G 1 L-modules is not closed under finite direct sums. In Theorem 3.9 we characterize Harada rings. In Section 4, for X 2 the class of cosingular modules, we associate the class d*x 2 with the property G 2 L. A module M is a G 2 L-module, if for every submodule N of M, there exists a direct summand K of M such that K is contained in N and the factor N/K is a cosingular module. We prove that every torsion-free Z-module is G 2 L. It is also proved that every R-module is G 2 L if and only if every R-module is a direct sum of an injective module and a cosingular module. In Theorem 4.11 we show that if R is a right hereditary ring, then every R-module is G 2 L if and only if Z(M) is injective for every R-module M. It is easily seen that the properties G 1 L and G 2 L are two generalizations of lifting modules and we have the following hierarchy: Lifting G 1 L G 2 L. 2 General Properties Like in [1], we define a class X of R-modules to be any collection of R-modules such that X contains a zero module and any module which is isomorphic to some module in X also belongs to X. A class X is called s-closed if every submodule of a module in X also belongs to X. Proposition 2.1 The following are equivalent for a module M: (i) M is in d*x. (ii) Every submodule A of M can be written as A = N S with N is a direct summand of M and S is in X.

3 Two generalizations of lifting modules 601 (iii) For every submodule A of M there exists a decomposition M = M 1 M 2 such that M 1 A and A M 2 is in X. Proof. Clear. Lemma 2.2 (see [1, Lemma 2.2 (iii)]) If X is any class of modules, then d*x is s-closed. Proposition 2.3 If X is any class of modules, then the following are equivalent: (i) Every R-module is in d*x. (ii) Every injective R-module is in d*x. Proof. By Lemma 2.2 and the fact that every module is contained in an injective module. Proposition 2.4 Let X be any class of modules. If M 1 is a module in d*x and M 2 is a semisimple module, then M = M 1 M 2 is in d*x. Proof. By [1, Proposition 2.3 (ii)]. Theorem 2.5 Let X be an s-closed class of modules such that X is closed under finite direct sums. Let M 1 and M 2 be two modules in d*x such that M i is M j -projective (i, j = 1, 2). Then M = M 1 M 2 belongs to d*x. Proof. Let A be a submodule of M. Consider the submodule M 1 (A + M 2 ) of M 1. Since M 1 is in d*x, there exists a decomposition M 1 = A 1 B 1 such that A 1 M 1 (A + M 2 ) and M 1 (A + M 2 ) B 1 = B 1 (A + M 2 ) belongs to X. Therefore M = M 1 M 2 = A 1 B 1 M 2 = A + (M 2 B 1 ). Since M 2 (A + B 1 ) M 2 and M 2 is in d*x, there exists a decomposition M 2 = A 2 B 2 such that A 2 M 2 (A+B 1 ) and B 2 (M 2 (A+B 1 )) = B 2 (A+B 1 ) belongs to X. Since A 2 A + B 1, we have M = A + B 1 M 2 = A + B 1 B 2. Since M = (A 1 A 2 ) (B 1 B 2 ) and A 1 A 2 is (B 1 B 2 )-projective, there exists A A such that M = A B 1 B 2 by [11, 41.14]. Since B 1 (A + B 2 ) and B 2 (A + B 1 ) are in X, [B 1 (A + B 2 )] [B 2 (A + B 1 )] is in X. But X is s-closed. Then A (B 1 B 2 ) is in X. Hence M is in d*x. Corollary 2.6 Let X be an s-closed class of modules such that X is closed under finite direct sums. Let M 1 and M 2 be two projective modules in d*x. Then M is in d*x.

4 602 N. O. Ertaş and R. Tribak Proof. By Theorem 2.5. A module M is called a duo module provided every submodule of M is fully invariant. Proposition 2.7 Let X be an s-closed class of modules such that X is closed under finite direct sums. Let M = M 1 M 2 be a duo module such that M 1 and M 2 are two modules in d*x. Then M belongs to d*x. Proof. Let A be a submodule of M. As A is fully invariant we have A = (A M 1 ) (A M 2 ). Since M 1 and M 2 are in d*x, there exist decompositions A M 1 = A 11 A 12 and A M 2 = A 21 A 22, where A 11 is a direct summand of M 1, A 21 is a direct summand of M 2 and A 12 and A 22 are in X. Then A 11 A 21 is a direct summand of M. Moreover, A 12 A 22 is a submodule of M belonging to X. Therefore M belongs to d*x. Proposition 2.8 Let X be an s-closed class of modules such that X is closed under factor modules. Let M be a module which belongs to d*x. (i) If N is a fully invariant submodule of M, then M/N belongs to d*x. (ii) If N be a submodule of M such that the sum of N with any nonzero direct summand of M is a direct summand of M, then M/N is in d*x. Proof. (i) Let A/N be a submodule of M/N. Since M belongs to d*x, there exists a decomposition M = K K such that K A and A/K belongs to X. By [7, Lemma 5.4], M/N = (K + N)/N (K + N)/N. But [A/N]/[(N + K)/N] = [A/K]/[(N + K)/K]. Then [A/N]/[(N + K)/N] belongs to X since X is closed under factor modules. Therefore M/N belongs to d*x. (ii) Let A be a submodule of M such that N A. If A X, then A/N X. Suppose that A X. Since M is in d*x, there exists a nonzero direct summand K of M such that K A and A/K is in X. By assumption, N + K is a direct summand of M. It is clear that (N + K)/N is also a direct summand of M/N. Since [A/K]/[(N + K)/K] = [A/N]/[(N + K)/N], [A/N]/[(N + K)/N] is also in X. It follows that M/N belongs to d*x. Theorem 2.9 Let X be an s-closed class of modules such that X is closed under factor modules. Let M = M 1 M 2 such that M 1 belongs to d*x and M 2 is M 1 -projective. Then M belongs to d*x if and only if for every submodule N of M such that N + M 1 M, there exists a direct summand K of M such that N/K belongs to X.

5 Two generalizations of lifting modules 603 Proof. ( ) Clear. ( ) Let N be a submodule of M such that N + M 1 = M. Since M 2 is M 1 - projective, there exists a submodule N N such that M = N M 1. Since M/N = M1, M/N belongs to d*x. Then there exists a direct summand K/N of M/N such that (N/N )/(K/N ) = N/K belongs to X. Clearly, K is also a direct summand of M. Then M belongs to d*x. 3 G 1 L-Modules A module M is called a G 1 L-module (or G 1 L for short), if for every submodule N of M there exists a direct summand K of M such that K N and N/K is a small R-module. It is clear that lifting modules and small modules are G 1 L. Since the class of small modules is closed under submodules, factor modules and finite direct sums, it follows that all the results of Section 2 are true for G 1 L-modules. Examples 3.1 (1) Let R be a commutative domain. Harada proved ([4, Theorem 2]) that the module R R is small. Therefore R R is a G 1 L-module. In particular, Z Z is a G 1 L-module. On the other hand, it is clear that Z Z is not lifting. (2) Let R be a Dedekind domain. Since every injective R-module is radical, every finitely generated R-module is small. Thus every finitely generated R- module is G 1 L. (3) Let M denote the Z-module Z/2Z Z/8Z. Then M is a G 1 L-module by (2). On the other hand, M is not lifting by [2, Example 22.5]. (4) It is easily seen from Proposition 2.4 that any module which is a direct sum of a semisimple module and a G 1 L-module is G 1 L. Then for every prime numbers p and q, the Z-module M = Z(p ) Z/qZ is a G 1 L-module. Note that M is neither lifting nor small. The following example shows that the condition M i is M j -projective (i, j = 1, 2) is not necessary in Theorem 2.5. Example 3.2 Let R be a commutative noetherian local ring with maximal ideal m and let I be an ideal of R such that I m and the ring R/I is a local ring with exactly one additional prime ideal, p/i, and such that the integral closure of R/p is also local. Consider the R-module M = R/m (R/I) p with (R/I) p is the total quotient ring of R/I (e.g. if R is a discrete valuation ring and p = I = 0, then (R/I) p = Q(R) the quotient field of R). It is clear by [6, Proposition 5.10] that (R/I) p is a hollow module. So it is a G 1 L-module. By

6 604 N. O. Ertaş and R. Tribak Proposition 2.4, M is a G 1 L-module. On the other hand, by [6, Proposition 5.10 and Lemma 5.11], R/m and (R/I) p are not relatively projective. Proposition 3.3 An indecomposable module M is G 1 L if and only if every proper submodule of M is a small module. Proof. Clear. A submodule L M is called coclosed in M, if for any proper submodule K L, there is a submodule N of M such that L + N = M but K + N M. A module M is called weakly injective if for every extension X of M, M is coclosed in X. It is known that every factor module of M is weakly injective if and only if M is non-cosingular (see [12]). Proposition 3.4 The following are equivalent for a weakly injective module M: (i) M is a G 1 L-module. (ii) M is lifting. Proof. By Proposition 2.1 and [2, 3.7(3) and 22.3]. Corollary 3.5 Let M be a module. If M is non-cosingular or injective, then M is G 1 L if and only if M is lifting. Proof. By Proposition 3.4. Corollary 3.6 An indecomposable injective module M is G 1 L if and only if M is hollow. Proof. Clear. Corollary 3.7 Let M be an injective Z-module. Then M is G 1 L if and only if M = i I [Z(p i )] (ni) where p i (i I) are primes and n i (i I) are natural numbers. Moreover, every submodule of M is G 1 L. Proof. By [6, Propositions A.7 and A.8], Proposition 3.4 and Lemma 2.2. In general, direct sums of G 1 L-modules is not G 1 L as seen by the following examples.

7 Two generalizations of lifting modules 605 Examples 3.8 (1) Let R be an incomplete rank one discrete valuation ring, with quotient field K. By [6, Lemma A.7], the module M = K 2 is not lifting. Since M is injective, M is not a G 1 L-module by Corollary 3.5. However, K is a G 1 L-module since K is a hollow module. (2) Let p be a prime number and let E = Z(p ) be the injective hull of the Z-module Z/pZ. By [2, 25.2], M = E (N) is not lifting. Therefore M is not G 1 L by Corollary 3.5. Theorem 3.9 The following are equivalent for a ring R: (i) Every R-module is a G 1 L-module. (ii) Every injective R-module is a G 1 L-module. (ii) R is a right Harada ring. Proof. (i) (ii) By Proposition 2.3. (ii) (iii) Let M be an R-module. Since E(M) is a G 1 L-module, M = N K such that N is a direct summand of E(M) and K is a small R-module by Proposition 2.1. We conclude from [2, 28.10] that R is a right Harada ring. (iii) (i) It is a consequence of [2, 28.10] and Proposition G 2 L-Modules Let M be a module. In [10], Talebi and Vanaja defined Z(M) = {U M M/U is small}. If Z(M) = 0 ( Z(M) = M), then M is called cosingular (non-cosingular). It is easy to check that every small module is cosingular. In this section, as a proper generalization of G 1 L-modules, we introduce the notion of G 2 L-modules. The module M is called a G 2 L-module (or G 2 L for short), provided for every submodule A of M there exists a direct summand K of M such that K A and A/K is cosingular. It is clear that G 1 L-modules and cosingular modules are G 2 L. In the following example we will give a counterexample to separate the properties G 1 L and G 2 L. Examples 4.1 (1) Let R be a Dedekind domain. Since every injective R- module is radical, every finitely generated R-module is small. Hence every finitely generated R-module is cosingular. Thus every direct sum of finitely generated R-modules is cosingular by [10, Corollary 2.2]. Therefore every direct sum of finitely generated R-modules is G 2 L. (2) Let R be a discrete valuation ring with maximal ideal m and quotient field K. Consider the R-module M = R (N). By (1), R R is a small module and hence Z(M) = 0 by [10, Proposition 2.1 (4)]. So M is a G 2 L-module.

8 606 N. O. Ertaş and R. Tribak Let r be any nonzero element of m and let N = Mr = (rr) (N). Suppose that M is a G 1 L-module. Then N = A B with A is a direct summand of M and B is a small module. Since R is a principal ideal ring, A is a direct sum of cyclic submodules of N. But every cyclic submodule of N is small in M. Then A = 0. Therefore N = B is a small R-module, a contradiction (see [5, Theorem 5]). Consequently, M is not a G 1 L-module. (3) If p is any prime number, then the Z-module M = Z(p ) is not cosingular ( Z(M) = M). On the other hand, M is a G 2 L-module since it is a hollow module. (4) For every module M, the factor module M/ Z(M) is a G 2 L-module since it is cosingular by [10, Proposition 2.1]. Lemma 4.2 (see [10, Corollary 2.2]) Let M be an R-module. The class of all cosingular modules is closed under submodules and direct sums. It follows immediately from the last lemma that all the results from 2.1 to 2.7 of Section 2 are true for G 2 L-modules. Lemma 4.3 Every non-cosingular submodule of a G 2 L-module M is a direct summand of M. Proof. Let N be a non-cosingular submodule of M. By Proposition 2.1, there exists a decomposition N = N 1 N 2, where N 1 is a direct summand of M and N 2 is cosingular. Since N is non-cosingular, we see that N 2 = 0 by [10, Proposition 2.1 (4)]. Hence N is a direct summand of M. Proposition 4.4 Let M be a G 2 L-module. Then Z 2 (M) = Z( Z(M)) is noncosingular and it is a direct summand of M. Proof. Since Z(M) is a submodule of M and M is a G 2 L-module, there exists a decomposition M = K K such that K Z(M) and Z(M)/K is cosingular. This gives Z 2 (M) + K = K by [10, Proposition 2.1 (1)]. Thus Z 2 (M) K. But Z 2 (M) = Z 2 (K) Z 2 (K ). Then Z 2 (M) = Z 2 (K). Since Z(M)/ Z 2 (M) is cosingular ([10, Proposition 2.1 (7)]), K/ Z 2 (M) is cosingular. Thus Z(K/ Z 2 (K)) = 0. It follows that Z(K) + Z 2 (K) = Z 2 (K). Therefore Z(K) = Z 2 (K) = Z 2 (M). Hence Z 2 (M) is non-cosingular. By Lemma 4.3, Z 2 (M) is a direct summand of M. Proposition 4.5 Let M be a non-cosingular amply supplemented module. Then M is G 2 L if and only if M is lifting.

9 Two generalizations of lifting modules 607 Proof. Assume that M is G 2 L. Let K be a coclosed submodule of M. By [10, Lemma 2.3 (3)], K is non-cosingular. Therefore K is a direct summand of M by Lemma 4.3. It follows from [2, 22.3] that M is lifting. The converse is clear. Proposition 4.6 Let M be an indecomposable module. Then M is G 2 L if and only if every proper submodule of M is cosingular. Proof. ( ) Let A be a proper submodule of M. By assumption, there exists a direct summand K of M such that K A and A/K is cosingular. Since M is indecomposable, we get K = 0. Hence A is cosingular. ( ) Follows from Proposition 2.1. Proposition 4.7 Let M be a non-cosingular G 2 L-module. Then every submodule of M is a direct sum of a non-cosingular module and a cosingular module. Proof. Let A be a submodule of M. By Proposition 2.1, A = A 1 A 2, where A 1 is direct summand of M and A 2 is cosingular. Clearly, A 1 is non-cosingular. This completes the proof. Lemma 4.8 Let M be an injective module. Then M is a G 2 L-module if and only if every submodule of M is a direct sum of an injective module and a cosingular module. Proof. ( ) By Proposition 2.1 and the fact that every direct summand of an injective module is again injective. ( ) Let L be a submodule of M. By assumption, L = L 1 L 2 such that L 1 is injective and L 2 is cosingular. Clearly, L/L 1 is cosingular and L 1 is a direct summand of M. So M is a G 2 L-module. It is easily seen from Theorem 3.9 that over Harada rings, every module is G 2 L. Theorem 4.9 The following are equivalent for a ring R: (i) Every R-module is G 2 L. (ii) Every injective R-module is G 2 L. (iii) Every R-module is a direct sum of an injective module and a cosingular module.

10 608 N. O. Ertaş and R. Tribak Proof. (i) (ii) By Proposition 2.3. (ii) (iii) Let M be a module and let E(M) be the injective hull of M. By assumption, E(M) is a G 2 L-module. By Lemma 4.8, M is a direct sum of an injective module and a cosingular module. (iii) (ii) By Lemma 4.8. An internal direct sum i I X i of submodules of a module M is called a local summand of M if, given any finite subset F of the index set I, the direct sum i F X i is a direct summand of M. Lemma 4.10 If every R-module is G 2 L, then every non-cosingular module is injective and it is a direct sum of indecomposabale modules. Proof. Let M be a non-cosingular module. Let X = i I X i be a local summand of M. Since all X i (i I) are non-cosingular, we have X i = Z(X i ) Z(X). Therefore X Z(X). Thus X is non-cosingular. By Theorem 4.9, X = A B where A is injective and B is cosingular. But B is non-cosingular by [10, Proposition 2.4]. Then B = 0 and X is injective. It follows that X is a direct summand of M. By [6, Theorem 2.17], M is a direct sum of indecomposable modules. Proposition 4.11 If R is a right hereditary ring, then the following conditions are equivalent: (i) Every R-module is G 2 L. (ii) R is right noetherian and for every R-module M, M is cosingular if and only if M has no nonzero injective submodules. (iii) For every R-module M, Z(M) is injective. (iv) Every non-cosingular R-module is injective and for every R-module M, Z(M) is a direct summand of M. If these conditions hold, then for every R-module M, Z(M) = {E M E is injective}. Proof. (i) (ii) Since R is right hereditary, the injective modules are exactly the non-cosingular modules by [10, Proposition 2.7] and Lemma Therefore every injective module is a direct sum of indecomposable modules by Lemma Thus R is noetherian by [9, Theorem 4.4]. The last statement follows from Theorem 4.9. (ii) (iii) Let M be an R-module. By [9, Exercise 4.1 p. 118], M = E F where E is an injective module which contains every injective submodule of M. So F is cosingular by assumption. On the other hand, E is non-cosingular by [10, Proposition 2.7]. Thus Z(M) = E is injective.

11 Two generalizations of lifting modules 609 (iii) (iv) Clear. (iv) (i) Let M be an R-module. We have M = Z(M) N for some submodule N of M. By [10, Proposition 2.1 (3)], N is cosingular. Therefore Z(M) = Z 2 (M). Thus Z(M) is non-cosingular. By hypothesis, Z(M) is injective. The result follows from Theorem 4.9. The last statement follows from (iii) and [10, Proposition 2.1 (1) and Proposition 2.7]. A module M over a Dedekind domain is called reduced if it has no nonzero divisible submodules. Lemma 4.12 Let M be a module over a Dedekind domain R. Then M is non-cosingular if and only if M is injective. Proof. Assume that M is non-cosingular. It is well known that M = E F such that E is injective and F is a reduced module. By [12, Satz 2.6], F is semisimple projective. Since every cyclic Z-module is small, Z(F ) = 0. But Z(F ) = F. Then F = 0 and M = E is injective. Conversely, it is clear that every injective R-module is non-cosingular by [10, Proposition 2.7]. Proposition 4.13 Let R be a Dedekind domain. The following are equivalent: (i) Every R-module is G 2 L. (ii) Every injective R-module is G 2 L. (iii) R is a field. Proof. (i) (iii) Let M be an R-module. By [13, Bemerkung 1.7 and Satz 2.10], there exists an R-module N such that M N and M = Z 2 (N). Since N is G 2 L, Proposition 4.4 shows that M is non-cosingular. Thus M is injective by Lemma Therefore every R-module is injective. It follows that R is semisimple. Thus R is a field. (iii) (ii) Clear. (ii) (i) See Theorem 4.9. Remarks 4.14 (1) The Z-module Z is a G 2 L-module but it follows easily from Proposition 4.13 that the ring Z has a module which is not G 2 L. (2) From Propositions 4.11, 4.13 and Lemma 4.12, it may be concluded that there exists a reduced Z-module which is not cosingular. Corollary 4.15 There is an injective Z-module M which is not G 2 L.

12 610 N. O. Ertaş and R. Tribak Proof. By Proposition Example 4.16 Let M be a Z-module which is not G 2 L (see Corollary 4.15). Let I be an index set such that M is a homomorphic image of Z (I). Since Z (I) is cosingular, Z (I) is G 2 L. This proves that in general the class of G 2 L-modules need not be closed under homomorphic images. Proposition 4.17 Let M = i I M i with M i = Q, the additive group of rational numbers, for all i I. Then every reduced submodule of M is cosingular. Proof. Let N be a reduced submodule of M. Assume that Z(N) 0. Let 0 x Z(N). Since Z(N) = s Z,s 0 sn by [13, Lemma 1.1 and Bemerkung 1.7], we have x sn for every nonzero element s Z. Consider the map ϕ : Q N defined by ϕ(r/s) = ry, where r Z, 0 s Z and y is an element of N such that x = sy. Note that ϕ is well defined since N is torsion-free. It is a simple matter to check that ϕ is a monomorphism. Thus Q = ϕ(q). So ϕ(q) is injective, a contradiction. Therefore Z(N) = 0. Corollary 4.18 Let M = i I M i with M i = Q, the additive group of rational numbers, for all i I. Then M is G 2 L but not G 1 L. Proof. It is well known that every Z-module is a direct sum of an injective module and a reduced module. The result follows from Lemma 4.8, Proposition 4.17, Corollary 3.5 and [2, 20.12]. Corollary 4.19 Every torsion-free Z-module is G 2 L. Proof. Let M be a torsion-free Z-module. Then E(M), the injective hull of M, is torsion-free. Thus E(M) = Q (I) for some index set I. By Corollary 4.18, E(M) is a G 2 L-module. Therefore M is G 2 L by Lemma 2.2. Remark 4.20 By Corollary 4.15, there exists an injective Z-module M such that M is not G 2 L. It is well known that M is a direct sum of modules each isomorphic to Q, the additive group of rational numbers, or to Z(p ) (for various primes p). Note that Q and Z(p ) are G 2 L (see Corollary 4.18). This proves that in general the class of G 2 L-modules need not be closed under direct sums.

13 Two generalizations of lifting modules 611 Let Mod-R denote the category of all right R-modules. For any R-module M, let σ[m] denote the full subcategory of Mod-R whose objects are isomorphic to a submodule of an M-generated module (see [11]). Recall (see [11, 23.1]) that a module M is called cosemisimple if each factor module of M has zero (Jacobson) radical and, for any ring R, the right R- module R R is cosemisimple precisely when every simple right R-module is injective, i.e., R is a right V -ring. Lemma 4.21 Let M be a cosemisimple module and let N be a module in σ[m]. If N is a G 2 L-module, then N is semisimple. Proof. Since M is cosemisimple, every module in σ[m] is non-cosingular by [10, Proposition 2.5]. It follows from Lemma 4.3 that every submodule of N is a direct summand of N. So N is semisimple. Corollary 4.22 Suppose that R is a right V -ring. Then every G 2 L-module over R is semisimple. Proof. By Lemma A ring R is a right max ring if every right module M 0 has at least one maximal submodule. Proposition 4.23 Let R be a ring such that every G 2 L-module is semisimple. Then R is a right max ring. Proof. Since every small module is a G 2 L-module, every small module is semisimple. It follows that for every module M, Rad(M) is semisimple. Let N be an R-module such that Rad(N) = N. Then N is semisimple, and hence Rad(N) = 0. Thus N = 0. Therefore for every nonzero module M, we have Rad(M) M. Consequently, R is a right max ring. A subset I of a ring R is called right T-nilpotent in case for every sequence a 1, a 2,... in I there is a natural number n such that a n... a 1 = 0. Corollary 4.24 Let R be a commutative ring such that every G 2 L-module is semisimple. Then R/J is a von Neumann regular ring and J = Rad(R) is T -nilpotent. Proof. By Proposition 4.23 and [3, First Max Theorem p. 203].

14 612 N. O. Ertaş and R. Tribak References [1] I. AL-Khazzi and P.F. Smith, Classes of modules with many direct summands, J. Aust. Math. Soc. (Series A), 59, (1995), [2] J. Clark, C. Lomp, N. Vanaja and R. Wisbauer, Lifting Modules, Frontiers in Mathematics, Birkhäuser, [3] Carl Faith, Rings whose modules have maximal submodules, Publ. Mat., 39, (1995), [4] Manabu Harada, On small submodules in the total quotient ring of a commutative ring, Rev. Un. Mat. Argentina, 28, (1977), [5] W.W. Leonard, Small modules, Proc. Amer. Math. Soc., 17, (1980), [6] S. H. Mohamed and B. J. Müller, Continuous and Discrete Modules, London Math. Soc. Lecture Notes Series, 147, Cambridge, [7] N. Orhan, D. K. Tütüncü and R. Tribak, On hollow-lifting modules, Taiwanese J. Math., 11 (2), (2007), [8] A. Ç. Özcan, Modules with small cyclic submodules in their injective hulls, Comm. Algebra, 30 (4), (2002), [9] D. W. Sharpe and P. Vamos, Injective Modules, Lecture in Pure Mathematics, University of Sheffield, [10] Y. Talebi and N. Vanaja, The torsion theory cogenerated by M-small modules, Comm. Algebra, 30 (3), (2002), [11] R. Wisbauer, Foundations of Module and Ring Theory, Gordon and Breach, Philadelphia, [12] H. Zöschinger, Schwach-injektive moduln, Period. Math. Hungar., 52 (2), (2006), [13] H. Zöschinger, Kosinguläre und kleine moduln, Comm. Algebra 33, (2005), Received: January, 2009