Research Article | Open Access

Volume 2008 |Article ID 469725 | https://doi.org/10.1155/2008/469725

Zhanmin Zhu, Xiaoxiang Zhang, "AGQP-Injective Modules", International Journal of Mathematics and Mathematical Sciences, vol. 2008, Article ID 469725, 7 pages, 2008. https://doi.org/10.1155/2008/469725

# AGQP-Injective Modules

Revised20 Apr 2008
Accepted20 Jun 2008
Published22 Dec 2008

#### 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  (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 , 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 . From , we know that GP-injective rings need not to be P-injective. Following , 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  generalized the concept of GP-injective rings to that of AGP-injective rings. According to , a ring is called right AGP-injective if, for any , there exist a positive integer and a left ideal such that and . In , 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 , 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 , if it is a homomorphic image of . Let = End(), following , 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 , 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  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  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  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  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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