AGQP-Injective Modules

Abstract

Let 𝑅 be a ring and let 𝑀 be a right 𝑅-module with 𝑆 = End(𝑀_𝑅). 𝑀 is called almost general quasi-principally injective (or AGQP-injective for short) if, for any 0≠𝑠∈𝑆, there exist a positive integer 𝑛 and a left ideal 𝑋_𝑠𝑛 of 𝑆 such that 𝑠^𝑛≠0 and 𝐥_𝑆(Ker(𝑠^𝑛))=𝑆𝑠^𝑛⊕𝑋_𝑠𝑛. 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 𝑅 is an associative ring with identity, and all modules are unitary. Recall that a ring 𝑅 is called right principally injective [1] (or right 𝑃-injective for short) if, every homomorphism from a principal right ideal of 𝑅 to 𝑅 can be extended to an endomorphism of 𝑅, or equivalently, 𝐥𝐫(𝑎)=𝑅𝑎 for all 𝑎∈𝑅. 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 𝑅 is called right GP-injective if, for any 0≠𝑎∈𝑅, there exists a positive integer 𝑛 such that 𝑎^𝑛≠0 and any right 𝑅-homomorphism from 𝑎^𝑛𝑅 to 𝑅 can be extended to an endomorphism of 𝑅. 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 𝑅-module 𝑀_𝑅 with 𝑆=End(𝑀_𝑅) is called quasiprincipally injective (or QP-injective for short) if, every homomorphism from an 𝑀-cyclic submodule of 𝑀 to 𝑀 can be extended to an endomorphism of 𝑀, or equivalently, 𝐥_𝑆(Ker(𝑠))=𝑆𝑠 for all 𝑠∈𝑆. In 1998, Page and Zhou [6] generalized the concept of GP-injective rings to that of AGP-injective rings. According to [6], a ring 𝑅 is called right AGP-injective if, for any 0≠𝑎∈𝑅, there exist a positive integer 𝑛 and a left ideal 𝑋_𝑎𝑛 such that 𝑎^𝑛≠0 and 𝐥𝐫(𝑎^𝑛)=𝑅𝑎^𝑛⊕𝑋_𝑎𝑛. 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 𝑅-module 𝑀 with 𝑆=End(𝑀_𝑅) is called GQP-injective if, for any 0≠𝑠∈𝑆, there exists a positive integer 𝑛 such that 𝑠^𝑛≠0 and any right 𝑅-homomorphism from 𝑠^𝑛(𝑀) to 𝑀 can be extended to an endomorphism of 𝑀, or equivalently, for any 0≠𝑠∈𝑆, there exists a positive integer 𝑛 such that 𝑠^𝑛≠0 and 𝐥_𝑆(Ker(𝑠^𝑛))=𝑆𝑠^𝑛. 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 𝑀_𝑅 be a right 𝑅-module with 𝑆=End(𝑀_𝑅). Then, 𝑀 is said to be almost general quasiprincipally injective (briefly, AGQP-injective) if, for any 0≠𝑠∈𝑆, there exist a positive integer 𝑛 and a left ideal 𝑋_𝑠𝑛 of 𝑆 such that 𝑠^𝑛≠0 and 𝐥_𝑆(Ker(𝑠^𝑛))=𝑆𝑠^𝑛⊕𝑋_𝑠𝑛.

Clearly, a ring 𝑅 is right AGP-injective if and only if 𝑅_𝑅 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 𝑀_𝑅 be a right 𝑅-module with 𝑆=End(𝑀_𝑅). Then,
(1)if 𝑆 is right AGP-injective, then 𝑀_𝑅 is AGQP-injective;(2)if 𝑀_𝑅 is AGQP-injective and 𝑀 generates Ker(𝑠) for each 𝑠∈𝑆, then 𝑆 is right AGP-injective.

Proof. (1) Suppose that 𝑆 is right AGP-injective then for any 0≠𝑠∈𝑆, there exist a positive integer 𝑛 and a left ideal 𝐼_𝑠𝑛 of 𝑆 such that 𝑠^𝑛≠0 and 𝐥_𝑆𝐫_𝑆(𝑠^𝑛)=𝑆𝑠^𝑛⊕𝐼_𝑠𝑛. If 𝑎∈𝐥_𝑆(Ker(𝑠^𝑛)) and 𝑏∈𝐫_𝑆(𝑠^𝑛), then 𝑠^𝑛𝑏=0, that is, 𝑏(𝑀)⊆Ker(𝑠^𝑛). Hence, (𝑎𝑏)𝑀=0, that is, 𝑎𝑏=0. This shows that 𝐥_𝑆(Ker(𝑠^𝑛))⊆𝐥_𝑆𝐫_𝑆(𝑠^𝑛). Therefore, we have 𝑆𝑠^𝑛⊆𝐥_𝑆(Ker(𝑠^𝑛))⊆𝑆𝑠^𝑛⊕𝐼_𝑠𝑛, which guarantees that𝐥_𝑆(Ker(𝑠^𝑛))=𝑆𝑠^𝑛⊕(𝐥_𝑆(Ker(𝑠^𝑛))∩𝐼_𝑠𝑛). (2.1)Thus, (1) is proved.
(2) Suppose that 𝑀_𝑅 is AGQP-injective then for any 0≠𝑠∈𝑆, there exist a positive integer 𝑛 and a left ideal 𝑋_𝑠𝑛 of 𝑆 such that 𝑠^𝑛≠0 and 𝐥_𝑆(Ker(𝑠^𝑛))=𝑆𝑠^𝑛⊕𝑋_𝑠𝑛. Assume that 𝑎∈𝐥_𝑆𝐫_𝑆(𝑠^𝑛) and Ker(𝑠^𝑛)=∑_𝑡∈𝑇𝑡(𝑀) for some subset 𝑇 of 𝑆. It is easy to see that 𝑎𝑡=0 for each 𝑡∈𝑇, so we have 𝑎𝑥=0 for each 𝑥∈Ker(𝑠^𝑛). This implies that 𝐥_𝑆𝐫_𝑆(𝑠^𝑛)⊆𝐥_𝑆(Ker(𝑠^𝑛)), from which we have𝑆𝑠^𝑛⊆𝐥_𝑆𝐫_𝑆(𝑠^𝑛)⊆𝐥_𝑆(Ker(𝑠^𝑛))=𝑆𝑠^𝑛⊕𝑋_𝑠𝑛, (2.2)and hence𝐥_𝑆𝐫_𝑆(𝑠^𝑛)=𝑆𝑠^𝑛⊕(𝐥_𝑆𝐫_𝑆(𝑠^𝑛)∩𝑋_𝑠𝑛). (2.3)Therefore, 𝑆 is right AGP-injective.

Recall that a module 𝑁 is called 𝑀-cyclic [3], if it is a homomorphic image of 𝑀. Let 𝑆 = End(𝑀_𝑅), following [8], we write 𝑊(𝑆)={𝑠∈𝑆∣Ker(𝑠)⊆^ess𝑀}.

Theorem 2.3. Let 𝑀_𝑅 be an AGQP-injective module with 𝑆=End(𝑀_𝑅). Then,
(1)𝑊(𝑆)⊆𝐽(𝑆),(2) if every nonzero submodule of 𝑀 contains a nonzero 𝑀-cyclic submodule, then 𝑊(𝑆)=𝐽(𝑆).

Proof. (1) Let 𝑠∈𝑊(𝑆). Then, for each 𝑡∈𝑆, 𝑡𝑠∈𝑊(𝑆) and so 1−𝑡𝑠≠0. Since 𝑀_𝑅 is AGQP-injective, there exist a positive integer 𝑛 and a left ideal 𝑋_{(1−𝑡𝑠)}^𝑛 such that (1−𝑡𝑠)^𝑛≠0 and 𝐥_𝑆(Ker(1−𝑡𝑠)^𝑛)=𝑆(1−𝑡𝑠)^𝑛⊕𝑋_{(1−𝑡𝑠)}^𝑛. Note that (1−𝑡𝑠)^𝑛=1−𝑢 for some 𝑢∈𝑊(𝑆). Since Ker(𝑢)∩Ker(1−𝑢)=0, we have Ker(1−𝑢)=0, and then 𝑆=𝑆(1−𝑢)⊕𝑋_1−𝑢. So 1=𝑒+𝑥 for some 𝑒∈𝑆(1−𝑢) and 𝑥∈𝑋_1−𝑢, it follows that 𝑒²=𝑒 and 𝑆(1−𝑢)=𝑆𝑒⊕𝑆(1−𝑒)∩𝑆(1−𝑢)=𝑆𝑒. Therefore, 1−𝑢=𝑣𝑒 for some 𝑣∈𝑆, since Ker(𝑢) is essential in 𝑀_𝑅, if 𝑒≠1, then there exists a nonzero element (1−𝑒)𝑚∈(1−𝑒)𝑀∩Ker(𝑢), and hence (1−𝑢)(1−𝑒)𝑚=(1−𝑒)𝑚. But (1−𝑢)(1−𝑒)𝑚=𝑣𝑒(1−𝑒)𝑚=0, a contradiction. So 𝑒=1, and hence 1−𝑢 is left invertible, which implies 𝑠∈𝐽(𝑆).
(2) We need only to prove that 𝐽(𝑆)⊆𝑊(𝑆). Let 𝑠∈𝐽(𝑆). If 𝑠∉𝑊(𝑆), then there exists 0≠𝑡∈𝑆 such that Ker(𝑠)∩𝑡(𝑀)=0 by hypothesis. Clearly, 𝑠𝑡≠0 and Ker(𝑠𝑡)=Ker(𝑡). Since 𝑀_𝑅 is AGQP-injective, there exist a positive integer 𝑛 and a left ideal 𝑋_(𝑠𝑡)^𝑛 such that (𝑠𝑡)^𝑛≠0 and𝐥_𝑆(Ker(𝑠𝑡)^𝑛)=𝑆(𝑠𝑡)^𝑛⊕𝑋_(𝑠𝑡)^𝑛. (2.4)If 𝑚∈Ker(𝑠𝑡)^𝑛, then (𝑠𝑡)^𝑛−1𝑚∈Ker(𝑠𝑡)=Ker(𝑡), and so 𝑚∈Ker(𝑡(𝑠𝑡)^𝑛−1). This shows that Ker(𝑠𝑡)^𝑛=Ker(𝑡(𝑠𝑡)^𝑛−1). Hence, 𝑡(𝑠𝑡)^𝑛−1∈𝑆(𝑠𝑡)^𝑛⊕𝑋_(𝑠𝑡)^𝑛. Write 𝑡(𝑠𝑡)^𝑛−1=𝑢(𝑠𝑡)^𝑛+𝑣, where 𝑢∈𝑆,𝑣∈𝑋_(𝑠𝑡)^𝑛. Then (1−𝑢𝑠)𝑡(𝑠𝑡)^𝑛−1=𝑣, which gives that (𝑠𝑡)^𝑛=𝑠(1−𝑢𝑠)⁻¹𝑣∈𝑆(𝑠𝑡)^𝑛∩𝑋_(𝑠𝑡)^𝑛=0, a contradiction.

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

Following [9], for a set 𝑋⊆ Hom (𝑁_𝑅,𝑀_𝑅), the submoduleKer𝑋=∩{Ker𝑔∣𝑔∈𝑋} (2.5) of 𝑁 is called an 𝑀-annihilator submodule of 𝑁. By [7, Lemma 9] and Theorem 2.3, we have the following corollary.

Corollary 2.5. Let 𝑀_𝑅 be an AGQP-injective module with 𝑆=End(𝑀_𝑅). If every nonzero submodule of 𝑀 contains a nonzero 𝑀-cyclic submodule, and 𝑀/Soc(𝑀) satisfies ACC on 𝑀-annihilator submodules, then 𝐽(𝑆) is nilpotent.

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

Theorem 2.6. Let 𝑀_𝑅 be an AGQP-injective module. Then,
(1)if 𝑀₁ and 𝑀₂ are submodules of 𝑀 such that 𝑀₁⊆𝑀₂ and 𝑀₁≅𝑀₂∣𝑀, then 𝑀₁∣𝑀. In particular 𝑀 is a GC2 module;(2)if 𝑀₁ and 𝑀₂ are simple submodules of 𝑀 such that 𝑀₁≅𝑀₂∣𝑀, then 𝑀₁∣𝑀.

Proof. (1) Let 𝑆=End(𝑀_𝑅). It is trivial in case 𝑀₁=0. Now suppose that 𝑀₁≠0 and 𝑀≅₂^𝑓𝑀₁. Then 𝑀₁=𝑎𝑀 and 𝑀₂=𝑒𝑀, where 𝑒²=𝑒∈𝑆 and 𝑎=𝑓𝑒. Since 𝑀_𝑅 is AGQP-injective, there exist a positive integer 𝑛 and a left ideal 𝑋_𝑎𝑛 such that 𝑎^𝑛≠0 and 𝐥_𝑆(Ker(𝑎^𝑛))=𝑆𝑎^𝑛⊕𝑋_𝑎𝑛. Let 𝑎⁰=𝑒, then 𝑓⁻¹(𝑎^𝑖+1𝑀)=𝑎^𝑖𝑀(𝑖=0,1,…,𝑛−1) since 𝑀₁⊆𝑀₂=𝑒𝑀. So we have𝑎^𝑖𝑀∣𝑎^𝑖−1𝑀⟺𝑓⁻¹(𝑎^𝑖+1𝑀)∣𝑓⁻¹(𝑎^𝑖𝑀)⟺𝑎^𝑖+1𝑀∣𝑎^𝑖𝑀(𝑖=1,…,𝑛−1). (2.6)Consequently, 𝑎𝑀∣𝑒𝑀⇔𝑎²𝑀∣𝑎𝑀⇔⋯⇔𝑎^𝑛𝑀∣𝑎^𝑛−1𝑀. Thus, to show 𝑎𝑀∣𝑀, it suffices to show that 𝑎^𝑛𝑀∣𝑀. Note that 𝑎|_𝑒𝑀∶𝑒𝑀→𝑒𝑀 is monic and 𝑎^𝑛(𝑚)=𝑎^𝑛(𝑒𝑚) for every 𝑚∈𝑀, 𝑒𝑀≅^𝑎𝑛𝑎^𝑛𝑀 and hence Ker(𝑎^𝑛) = Ker(𝑒). It follows that 𝑒∈𝐥_𝑆(Ker(𝑒))=𝐥_𝑆(Ker(𝑎^𝑛))=𝑆𝑎^𝑛⊕𝑋_𝑎𝑛. Now, let 𝑒=𝑏𝑎^𝑛+𝑥 with 𝑏∈𝑆 and 𝑥∈𝑋_𝑎𝑛, then 𝑎^𝑛=𝑎^𝑛𝑒=𝑎^𝑛𝑏𝑎^𝑛+𝑎^𝑛𝑥=𝑎^𝑛𝑏𝑎^𝑛. Finally, let 𝑔=𝑎^𝑛𝑏, then 𝑔²=𝑔 and 𝑎^𝑛𝑀=𝑔𝑀 as required.
(2) Let 𝑀₂=𝑒₁𝑀, where 𝑒²₁=𝑒₁∈𝑆, and let 𝑀≅₂^𝑓1𝑀₁. Then 𝑀₁=𝑎₁𝑀, where 𝑎₁=𝑓₁𝑒₁. Since 𝑀_𝑅 is AGQP-injective, there exist a positive integer 𝑛₁ and a left ideal 𝑋_𝑎^𝑛₁₁ such that 𝑎^𝑛1₁≠0 and 𝐥_𝑆(Ker(𝑎^𝑛1₁))=𝑆𝑎^𝑛1₁⊕𝑋_𝑎^𝑛₁₁. Note that 0≠𝑎^𝑛1₁𝑀⊆𝑎₁𝑀, and 𝑎₁𝑀 is simple. We have 𝑎^𝑛1₁𝑀=𝑎₁𝑀. Clearly, Ker(𝑒₁)=Ker(𝑎₁) because 𝑓₁ is a monomorphism. Since 𝑎₁𝑀 is simple, Ker(𝑎₁) is a maximal submodule of 𝑀. But Ker(𝑎₁)⊆Ker(𝑎^𝑛1₁)≠𝑀, so Ker(𝑎₁)=Ker(𝑎^𝑛1₁) and then Ker(𝑒₁)=Ker(𝑎^𝑛1₁). It follows that 𝑒₁∈𝐥_𝑆(Ker(𝑒₁))=𝐥_𝑆(Ker(𝑎^𝑛1₁))=𝑆𝑎^𝑛1₁⊕𝑋_𝑎^𝑛₁₁. Now, let 𝑒₁=𝑏₁𝑎^𝑛1₁+𝑦 with 𝑏₁∈𝑆 and 𝑦∈𝑋_𝑎^𝑛₁₁, then 𝑎^𝑛1₁=𝑎^𝑛1₁𝑒₁=𝑎^𝑛1₁𝑏₁𝑎^𝑛1₁+𝑎^𝑛1₁𝑦=𝑎^𝑛1₁𝑏₁𝑎^𝑛1₁. Finally, let 𝑔₁=𝑎^𝑛1₁𝑏₁, then 𝑔²₁=𝑔₁ and 𝑀₁=𝑎₁𝑀=𝑎^𝑛1₁𝑀=𝑔₁𝑀 as required.

Recall that a module 𝑀 is said to be weakly injective [11] if, for any finitely generated submodule 𝑁≤𝐸(𝑀), there exists 𝑋≤𝐸(𝑀) such that 𝑁⊆𝑋≅𝑀.

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

Proof. We need only to prove the sufficiency. Let 𝑥∈𝐸(𝑀). Then, there exists 𝑋⊆𝐸(𝑀) such that 𝑀+𝑥𝑅⊆𝑋≅𝑀. Hence, 𝑋 is AGQP-injective and 𝑀∣𝑋 follows from Theorem 2.6(1). But 𝑀 is essential in 𝐸(𝑀), so 𝑀=𝑋 and hence 𝑥∈𝑀.

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

Proof. (1) Since 𝑀_𝑅 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 𝑀 and 𝑈 are two right 𝑅-modules, then 𝑈 is called M-projective in case for each epimorphism 𝑔∶𝑀_𝑅→𝑁_𝑅 and each homomorphism 𝛾∶𝑈_𝑅→𝑁_𝑅, there is an 𝑅-homomorphism 𝛾∶𝑈_𝑅→𝑀_𝑅 such that 𝛾=𝑔𝛾. A module 𝑀_𝑅 is called quasiprojective if it is 𝑀-projective.

Let 𝑅 be a ring. Recall that an element 𝑎∈𝑅 is called 𝜋-regular if there exists a positive integer 𝑚 such that 𝑎^𝑚=𝑎^𝑚𝑏𝑎^𝑚 [13] for some 𝑏∈𝑅. An element 𝑥∈𝑅 is called generalized 𝜋-regular if there exists a positive integer 𝑛 such that 𝑥^𝑛=𝑥^𝑛𝑦𝑥 for some 𝑦∈𝑅. A ring 𝑅 is called 𝜋-regular (resp., generalized 𝜋-regular) if every element in 𝑅 is 𝜋-regular (resp., generalized 𝜋-regular). If 𝐴 is a subset of 𝑅, then we say that 𝐴 is regular if every element in 𝐴 is regular.

Proposition 2.9. Let 𝑀_𝑅 be quasiprojective with 𝑆=End(𝑀_𝑅). Then, 𝑆 is regular if and only if 𝑀_𝑅 is AGQP-injective and 𝑠(𝑀) is 𝑀-projective for every 𝑠∈𝑆.

Proof. Assume that 𝑆 is regular. Then, every right ideal of 𝑆 is a direct summand of 𝑆_𝑆, and so every homomorphism from a principal right ideal of 𝑆 to 𝑆 can be extended to an endomorphism of 𝑆. Hence, 𝑆 is right P-injective and then right AGP-injective. By Theorem 2.2, 𝑀_𝑅 is AGQP-injective. The regularity of 𝑆 also implies that 𝑠(𝑀) is a direct summand of 𝑀 by [14, Theorem 37.7]. But 𝑀 is quasiprojective, so 𝑠(𝑀) is 𝑀-projective for every 𝑠∈𝑆.
Conversely, suppose 𝑀_𝑅 is AGQP-injective and 𝑠(𝑀) is 𝑀-projective for every 𝑠∈𝑆. Then for any 0≠𝑎∈𝑆, by the AGQP-injectivity of 𝑀_𝑅, there exist a positive integer 𝑛 and a left ideal 𝑋_𝑎𝑛 of 𝑆 such that 𝑎^𝑛≠0 and 𝐥_𝑆(Ker(𝑎^𝑛))=𝑆𝑎^𝑛⊕𝑋_𝑎𝑛. Since 𝑎^𝑛𝑀 is 𝑀-projective, Ker(𝑎^𝑛) = 𝑒𝑀 for some 𝑒²=𝑒∈𝑆. Then, we have 𝑆(1−𝑒)=𝐥_𝑆(𝑒𝑀)=𝐥_𝑆(Ker(𝑎^𝑛))=𝑆𝑎^𝑛⊕𝑋_𝑎𝑛, and so 1−𝑒=𝑏𝑎^𝑛+𝑥 for some 𝑏∈𝑆 and 𝑥∈𝑋_𝑎𝑛. Thus, 𝑎^𝑛=𝑎^𝑛(1−𝑒)=𝑎^𝑛𝑏𝑎^𝑛+𝑎^𝑛𝑥=𝑎^𝑛𝑏𝑎^𝑛. This proves that 𝑆 is 𝜋-regular and hence generalized 𝜋-regular. Clearly, 𝑁₁(𝑆)={0≠𝑎∈𝑆∣𝑎²=0} is regular (in this case, 𝑛 must be equal to 1). Therefore or, 𝑆 is regular by [13, Theorem 2.2].

Recall that a module 𝑀_𝑅 is called an IN-module [15] if 𝐥_𝑆(𝐴∩𝐵)=𝐥_𝑆(𝐴)+𝐥_𝑆(𝐵) for any submodules 𝐴 and 𝐵 of 𝑀, where 𝑆=End(𝑀_𝑅).

Proposition 2.10. Let 𝑀_𝑅 be an AGQP-injective IN-module with 𝑆=End(𝑀_𝑅). Then, 𝑆 is regular if and only if 𝑊(𝑆)=0.

Proof. By Theorem 2.3, we need only to prove the sufficiency. Let 0≠𝑎∈𝑆. Since 𝑀_𝑅 is AGQP-injective, there exist a positive integer 𝑛 and a left ideal 𝑋_𝑎𝑛 of 𝑆 such that 𝑎^𝑛≠0 and 𝐥_𝑆(Ker(𝑎^𝑛))=𝑆𝑎^𝑛⊕𝑋_𝑎𝑛. Since 𝑊(𝑆)=0, Ker(𝑎^𝑛) is not essential in 𝑀 and then there exists a nonzero submodule 𝐾 such that Ker (𝑎^𝑛)⊕𝐾 is essential in 𝑀. Moveover, we also have𝐥_𝑆(Ker(𝑎^𝑛))+𝐥_𝑆(𝐾)=𝐥_𝑆(Ker(𝑎^𝑛𝐥)∩𝐾)=𝑆,_𝑆(Ker(𝑎^𝑛))∩𝐥_𝑆(𝐾)⊆𝐥_𝑆(Ker(𝑎^𝑛)+𝐾)=0, (2.7)because 𝑀_𝑅 is an IN-module and 𝑊(𝑆)=0. Thus,𝑆=𝐥_𝑆(Ker(𝑎^𝑛))⊕𝐥_𝑆(𝐾)=𝑆𝑎^𝑛⊕𝑋_𝑎𝑛⊕𝐥_𝑆(𝐾). (2.8)Let 1=𝑏𝑎^𝑛+𝑥 with 𝑏∈𝑆, 𝑥∈𝑋_𝑎𝑛⊕𝐥_𝑆(𝐾), then 𝑎^𝑛=𝑎^𝑛𝑏𝑎^𝑛. It follows that 𝑆 is regular by the last part of the proof of Proposition 2.9.

Lemma 2.11. Let 𝑀_𝑅 be an AGQP-injective module in which every nonzero submodule contains a nonzero 𝑀-cyclic submodule and 𝑆=End(𝑀_𝑅). If 𝑠∉𝑊(𝑆), then the inclusion Ker(𝑠)⊆Ker(𝑠−𝑠𝑡𝑠) is strict for some 𝑡∈𝑆.

Proof. If 𝑠∉𝑊(𝑆), then Ker(𝑠)∩𝐾=0 for some nonzero submodule 𝐾 of 𝑀, and so Ker(𝑠)∩𝑠^′(𝑀)=0 for some 0≠𝑠^′∈𝑆 by hypothesis. Clearly, 𝑠𝑠^′≠0. Since 𝑀_𝑅 is AGQP-injective, there exist a positive integer 𝑛 and a left ideal 𝑋_{(𝑠𝑠′)}^𝑛 such that (𝑠𝑠^′)^𝑛≠0 and 𝐥_𝑆(Ker(𝑠𝑠^′)^𝑛)=𝑆(𝑠𝑠^′)^𝑛⊕𝑋_{(𝑠𝑠′)}^𝑛. Thus,𝑠^′(𝑠𝑠^′)^𝑛−1∈𝐥_𝑆(Ker(𝑠^′(𝑠𝑠^′)^𝑛−1)=𝐥_𝑆(Ker(𝑠𝑠^′)^𝑛)=𝑆(𝑠𝑠^′)^𝑛⊕𝑋_{(𝑠𝑠′)}^𝑛. (2.9)Write 𝑠^′(𝑠𝑠^′)^𝑛−1=𝑡(𝑠𝑠^′)^𝑛+𝑥, where 𝑡∈𝑆 and 𝑥∈𝑋_{(𝑠𝑠′)}^𝑛, then (1−𝑡𝑠)𝑠^′(𝑠𝑠^′)^𝑛−1=𝑥 and hence(1−𝑠𝑡)(𝑠𝑠^′)^𝑛=(𝑠−𝑠𝑡𝑠)𝑠^′(𝑠𝑠^′)^𝑛−1=𝑠𝑥∈𝑆(𝑠𝑠^′)^𝑛∩𝑋_{(𝑠𝑠′)}^𝑛. (2.10)This means that (𝑠−𝑠𝑡𝑠)𝑠^′(𝑠𝑠^′)^𝑛−1=0. It is obvious that Ker (𝑠)⊆ Ker(𝑠−𝑠𝑡𝑠). Note that 𝑠^′(𝑠𝑠^′)^𝑛−1𝑀 is contained in Ker (𝑠−𝑠𝑡𝑠) but not contained in Ker(𝑠), the inclusion Ker (𝑠)⊆ Ker (𝑠−𝑠𝑡𝑠) is strict.

Theorem 2.12. Let 𝑀_𝑅 be AGQP-injective with 𝑆=End(𝑀_𝑅). If every nonzero submodule of 𝑀 contains a nonzero 𝑀-cyclic submodule, then the following conditions are equivalent:
(1)𝑆 is right perfect;(2) for any sequence {𝑠₁,𝑠₂,…}⊆𝑆, the chain Ker(𝑠₁)⊆Ker(𝑠₂𝑠₁)⊆⋯ 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.