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International Journal of Mathematics and Mathematical Sciences
VolumeΒ 2008, Article IDΒ 469725, 7 pages
http://dx.doi.org/10.1155/2008/469725
Research Article

AGQP-Injective Modules

1Department of Mathematics, Jiaxing University, Jiaxing, Zhejiang 314001, China
2Department of Mathematics, Southeast University, Nanjing 210096, China

Received 23 December 2007; Revised 20 April 2008; Accepted 20 June 2008

Academic Editor: RobertΒ Lowen

Copyright Β© 2008 Zhanmin Zhu and Xiaoxiang Zhang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let be a ring and let be a right -module with = End(). is called almost general quasi-principally injective (or AGQP-injective for short) if, for any , there exist a positive integer and a left ideal of such that and . Some characterizations and properties of AGQP-injective modules are given, and some properties of AGQP-injective modules with additional conditions are studied.

1. Introduction

Throughout is an associative ring with identity, and all modules are unitary. Recall that a ring is called right principally injective [1] (or right -injective for short) if, every homomorphism from a principal right ideal of to can be extended to an endomorphism of , or equivalently, for all . The concept of right P-injective rings has been generalized by many authors. For example, in [2, 3], right P-injective rings are generalized in two directions, respectively. Following [2], a ring is called right GP-injective if, for any , there exists a positive integer such that and any right -homomorphism from to can be extended to an endomorphism of . Note that GP-injective rings are also called YJ-injective in [4]. From [5], we know that GP-injective rings need not to be P-injective. Following [3], a right -module with is called quasiprincipally injective (or QP-injective for short) if, every homomorphism from an -cyclic submodule of to can be extended to an endomorphism of or equivalently, for all . In 1998, Page and Zhou [6] generalized the concept of GP-injective rings to that of AGP-injective rings. According to [6], a ring is called right AGP-injective if, for any , there exist a positive integer and a left ideal such that and . In [7], the first author introduced the notion of GQP-injective modules which can be regarded as the generalization of GP-injective rings and QP-injective modules. According to [7], a right -module with is called GQP-injective if, for any , there exists a positive integer such that and any right -homomorphism from to can be extended to an endomorphism of , or equivalently, for any , there exists a positive integer such that and . The nice structure of AGP-injective rings and GQP-injective modules draws our attention to define almost GQP-injective modules, in a similar way to AGP-injective rings, and to investigate their properties.

2. Results

Definition 2.1. Let be a right -module with . Then, is said to be almost general quasiprincipally injective (briefly, AGQP-injective) if, for any , there exist a positive integer and a left ideal of such that and .

Clearly, a ring is right AGP-injective if and only if is AGQP-injective, GQP-injective modules are AGQP-injective.

Our next result gives the relationship between the AGQP-injectivity of a module and the AGP-injectivity of its endomorphism ring.

Theorem 2.2. Let be a right -module with . Then,
(1)if is right AGP-injective, then is AGQP-injective;(2)if is AGQP-injective and generates for each , then is right AGP-injective.

Proof. (1) Suppose that is right AGP-injective then for any , there exist a positive integer and a left ideal of such that and . If and , then , that is, . Hence, , that is, . This shows that . Therefore, we have , which guarantees thatThus, (1) is proved.
(2) Suppose that is AGQP-injective then for any , there exist a positive integer and a left ideal of such that and . Assume that and for some subset of . It is easy to see that for each , so we have for each . This implies that , from which we haveand henceTherefore, is right AGP-injective.

Recall that a module is called -cyclic [3], if it is a homomorphic image of . Let = End(), following [8], we write .

Theorem 2.3. Let be an AGQP-injective module with . Then,
(1),(2) if every nonzero submodule of contains a nonzero -cyclic submodule, then .

Proof. (1) Let . Then, for each , and so . Since is AGQP-injective, there exist a positive integer and a left ideal such that and . Note that for some . Since , we have , and then . So for some and , it follows that and . Therefore, for some , since is essential in , if , then there exists a nonzero element , and hence . But , a contradiction. So and hence is left invertible, which implies .
(2) We need only to prove that . Let . If , then there exists such that by hypothesis. Clearly, and . Since is AGQP-injective, there exist a positive integer and a left ideal such that andIf , then , and so . This shows that . Hence, . Write , where . Then , which gives that , a contradiction.

Corollary 2.4 (see [6, Corollary 2.3]). If is a right AGP-injective ring, then .

Following [9], for a set Hom , the submodule of is called an -annihilator submodule of . By [7, Lemma 9] and Theorem 2.3, we have the following corollary.

Corollary 2.5. Let be an AGQP-injective module with . If every nonzero submodule of contains a nonzero -cyclic submodule, and satisfies ACC on -annihilator submodules, then is nilpotent.

Recall that a module is said to be a GC2 module [10] if every submodule with is a direct summand of . For convenience, we write to denote that is a direct summand of .

Theorem 2.6. Let be an AGQP-injective module. Then,
(1)if and are submodules of such that and , then . In particular is a GC2 module;(2)if and are simple submodules of such that , then .

Proof. (1) Let . It is trivial in case . Now suppose that and . Then and , where and . Since is AGQP-injective, there exist a positive integer and a left ideal such that and . Let , then since . So we haveConsequently, . Thus, to show , it suffices to show that . Note that is monic and for every , and hence Ker() = Ker(). It follows that . Now, let with and , then . Finally, let , then and as required.
(2) Let , where , and let . Then , where . Since is AGQP-injective, there exist a positive integer and a left ideal such that and . Note that and is simple. We have . Clearly, because is a monomorphism. Since is simple, is a maximal submodule of . But , so and then . It follows that . Now, let with and , then . Finally, let , then and as required.

Recall that a module is said to be weakly injective [11] if, for any finitely generated submodule , there exists such that .

Corollary 2.7. Let be a finitely generated module. Then, is injective if and only if is weakly injective and AGQP-injective. In particular, a ring is right self-injective if and only if is weakly injective and AGP-injective.

Proof. We need only to prove the sufficiency. Let . Then, there exists such that . Hence, is AGQP-injective and follows from Theorem 2.6(1). But is essential in , so and hence .

Corollary 2.8. Let be an AGQP-injective module with .
(1)If is of finite Goldie dimension, then S is semilocal.(2)If is a noetherian self-generator, then is semiprimary.

Proof. (1) Since is AGQP-injective, it satisfies the GC2-condition by Theorem2.6(1) and then (1) follows immediately by [12, Lemma 1.1].
(2) By (1) and Corollary 2.5.

Recall that if and are two right -modules, then is called M-projective in case for each epimorphism and each homomorphism , there is an -homomorphism such that . A module is called quasiprojective if it is -projective.

Let be a ring. Recall that an element is called -regular if there exists a positive integer such that [13] for some . An element is called generalized -regular if there exists a positive integer such that for some . A ring is called -regular (resp., generalized -regular) if every element in is -regular (resp., generalized -regular). If is a subset of , then we say that is regular if every element in is regular.

Proposition 2.9. Let be quasiprojective with . Then, is regular if and only if is AGQP-injective and is -projective for every .

Proof. Assume that is regular. Then, every right ideal of is a direct summand of , and so every homomorphism from a principal right ideal of to can be extended to an endomorphism of . Hence, is right P-injective and then right AGP-injective. By Theorem 2.2, is AGQP-injective. The regularity of also implies that is a direct summand of by [14, Theorem 37.7]. But is quasiprojective, so is -projective for every .
Conversely, suppose is AGQP-injective and is -projective for every . Then for any , by the AGQP-injectivity of , there exist a positive integer and a left ideal of such that and . Since is -projective, Ker() = for some . Then, we have , and so for some and . Thus, . This proves that is -regular and hence generalized -regular. Clearly, is regular (in this case, must be equal to 1). Therefore or, is regular by [13, Theorem 2.2].

Recall that a module is called an IN-module [15] if for any submodules and of , where .

Proposition 2.10. Let be an AGQP-injective IN-module with . Then, is regular if and only if .

Proof. By Theorem 2.3, we need only to prove the sufficiency. Let . Since is AGQP-injective, there exist a positive integer and a left ideal of such that and . Since , Ker() is not essential in and then there exists a nonzero submodule such that Ker is essential in . Moveover, we also havebecause is an IN-module and . Thus,Let with , , then . It follows that is regular by the last part of the proof of Proposition 2.9.

Lemma 2.11. Let be an AGQP-injective module in which every nonzero submodule contains a nonzero -cyclic submodule and . If , then the inclusion is strict for some .

Proof. If , then Ker for some nonzero submodule of , and so Ker for some by hypothesis. Clearly, . Since is AGQP-injective, there exist a positive integer and a left ideal such that and . Thus,Write where and , then and henceThis means that . It is obvious that Ker Ker(). Note that is contained in Ker but not contained in Ker(), the inclusion Ker Ker is strict.

Theorem 2.12. Let be AGQP-injective with . If every nonzero submodule of contains a nonzero -cyclic submodule, then the following conditions are equivalent:
(1) is right perfect;(2) for any sequence , the chain terminates.

Proof. By Theorem 2.3, Lemma 2.11, and [16, Lemma 2.8], one can complete the proof in a similar way to that of [16, Theorem 2.9].

Acknowledgment

The authors are very grateful to the referees for their useful comments and suggestions.

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