Some Variations of -Supplemented Modules with Regard to a Hereditary Torsion Theory

Abstract

In present work, we describe and investigate torsion theoretic versions of -supplemented modules via a hereditary torsion theory . With this aim, first, we define -small submodules. On this basis, the concepts of -lifting modules, -supplemented modules, and amply -supplemented modules and their fundamental properties are given, respectively. Furthermore, we present -semiperfect modules and give a characterization for them via (amply) -supplemented modules. Even we supply binary relations between these new module classes.

1. Introduction

Along this work, an associative ring with a unit is denoted by , a unitary left -module is denoted by , and - is the category of unitary left -modules. The symbols “ and ” will denote a submodule and a direct summand of a module, respectively.

Let us point a community of modules with . The reject of in is described by . The module homomorphism satisfies . Whenever is onto and , is confirmed [1].

A submodule of is called small in (denoted by ) if for every proper submodule of . A submodule of is called essential in (denoted by ) if the intersection of with each submodule of is nonzero excluding . The community of elements of whose annihilators are essential in is described as the singular submodule of (denoted by ). is said to be singular (nonsingular) whenever [2]. A form of small submodules via singularity was contributed to the literature in [3] by Zhou. For a module , is said to be -small in (denoted by ) in case with singular implies that . Let be the community of whole singular simple modules. As it is indicated in [3], . For , a -supplement submodule of provides and . A -supplemented module is a module in which each submodule of is of a -supplement. Besides, is said to have ample-supplements in if every submodule of with involves a -supplement of in . An amply -supplemented module is a module in which each submodule of is of ample -supplements. Even is called -lifting if for each , there exists a decomposition of such that with and . is called distributive if for , the statement is verified. If for each , , we say is a fully invariant submodule of . We refer the interested readers to [4–6] for concepts given here.

Now, we give place to fundamental concepts of torsion theory. Let be a torsion theory on -, where denotes the community of all modules which are -torsion and denotes the community of all modules which are free of -torsion, that is, and such that . Ordinarily, is preserved under homomorphic images, extensions, and direct sums. In response to this, is preserved by isomorphisms, submodules, extensions, and direct products. If is preserved by submodules (injective hulls), then is called a hereditary (stable) torsion theory. In the present study, we will accept that is a hereditary torsion theory unless otherwise specified. A submodule of is defined as -dense (-pure) if is -torsion (-torsion free), denoted by (). For further properties associated with the torsion theory, we refer to [7].

In recent years, it is a lifting trend for algebraists to get torsion theoretic forms of known concepts or theories from ring and module theory. In [8], the authors handled lifting modules according to a (hereditary) torsion theory. In 1985, Pardo defined -essential submodules [9]. By using this fact, in 2017, the authors investigated singular and nonsingular modules according to a hereditary torsion theory to determine the structure of -extending modules [10], first defined in [11] according to Bland’s -essential submodules. They defined the set . The submodule is called a -singular submodule of . is called a -singular module provided , and is called a non--singular module provided . Furthermore, in [12], -complement submodules of a module are defined as a torsion theoretic version of complement submodules. Dually, supplemented modules, some generalizations, and characterizations of them are handled from this aspect by various authors [13].

In the present study, the structure of -supplemented modules is researched by using the concept of -singularity of a submodule according to Pardo’s -essential submodules. Motivated by this idea, we handle the special form of lifting modules given in [8] with respect to -singularity. To obtain this, first, we define -small submodules and give fundamental properties similar to -small submodules. In the light of this fact, we introduce -lifting, -supplemented, and amply -supplemented modules. We also interested in binary relations between these modules. Moreover, -semiperfect modules are presented, and characterizations of a -semiperfect module are given in view of being (amply) -supplemented under special conditions.

2. -Small Submodules

Definition 1. Let be a module and . If whenever is -singular for any , then is said to be -small in . The notation is prefered to point that is a -small submodule of .
Explicitly, each small submodule of a module is -small. Also, note that as -singular module classes are different from singular ones, there is not a certain relation between -small submodules and -small submodules. But if and are free of -torsion, then these concepts coincide.
Following lemma is given for a submodule of a module to be -small.

Lemma 2. For a module , the listed statements taking place below are equivalent:(1)(2)If  , then for a non--singular submodule with

Proof. Let . In this case, there subsists a submodule of maximal according to the feature . Thus, we obtain that by [12], Proposition 2.9. Following we have is -singular by [10], Theorem 3.7. Since and , we have . Let . Then, . Applying the same way as above by replacing with , we get . Thus, that verifies is semisample. So, we can write , where is non--singular. Then, is -singular. Since and , we have . This shows that ; that is, is non--singular.
: let for a submodule of with -singular. By hypothesis, there subsists with is non--singular. This shows that .
Now, we list the main features of -small submodules in the lemma mentioned as follows.

Lemma 3. The following statements given hold for a module .(1)For submodules , , and of with , we have(a) and (b) and (2)If and is a homomorphism, then . Most particularly, if , then .(3)Let , , and . Then, and .

Proof. The proofs can be repeated by a similar approach given for small submodules in ([1], 19.3).

Definition 4. Let be the community of whole -singular simple modules. For a module , let be the reject of in . If does not have any submodule with this type, then we denote .
It is an easy fact that .
We give a relation between -radical of a module and its -small submodules in the following lemma.

Lemma 5. Let be a module. Then we have, for any module, holds .

Proof. Let . We will show that is contained in every maximal submodule of with -singular. Assume that for a maximal submodule of with -singular. Then, since is maximal, . Then, , which is a contradiction to the fact that T is maximal in W. Hence, . For any , clearly is the element of all maximal submodules of with being -singular. Now, we claim that . Assume that is not -small in and . It is clear that , since is not -small in . By the Zorn Lemma, there exists a maximal element in . Accordingly, and so we have the contradiction . Hence, .
Now, we give some facts about -radical of a module.

Lemma 6. (1)If is a homomorphism, then . So is fully invariant.(2)If , then .(3) is the unique largest -small submodule of if every submodule of is contained in a maximal submodule of .

Proof. The proof can be repeated alike given in [1].

Corollary 7. If is a stable torsion theory or is free of -torsion, each -small submodule is -small in by ([10], Lemma 3.1).

As it is understood from the definitions, -small submodules need not be -small and the converse is also. They are only specialized versions of each other.

3. -Lifting Modules

In this department of the article, we introduce -lifting modules and present fundamental properties of them. First, we give matching conditions for a module to be -lifting, and afterwards, we handle the other structure theorems for homomorphic images, direct summands, direct sums, etc.

Definition 8. A module is called -lifting if for there exists a decomposition such that and .
If is -torsion free or is a stable torsion theory, then the case of being -lifting satisfies the case of being -lifting for a module since . Even these new concepts coincide for -torsion free modules over -torsion free rings since .
In the following theorem, we list the equivalent conditions for a module to be -lifting.

Theorem 9. (1)The following statements given are equivalent for a module :(a) is -lifting(b)For each , there exists submodules , providing , , and (c)For each , there exists , providing and (2)Every direct summand of a -lifting module inherits the property.

Proof. (1) It is obvious. Let . By hypothesis, there exists a decomposition of providing with and . For the natural epimorphism , we have , since . Let be any submodule of . By , there exists a decomposition of , providing with and . Therefore, and . Since and , then we get . Hence, is a -lifting module.(2)Let be -lifting and . In that case, there exists with . For any , since is -lifting, there exists a decomposition of providing with and . Therefore, it is obtained that providing and so as . Hence, because .The following example includes a -lifting module.

Example 1. Let be a matrix ring in which elements are upper triangular matrices with the form and components coming from the field , and , which is an idempotent ideal of . Here, is a hereditary torsion theory with the torsion part -. Let us list the all proper submodules of as follows: , , , and . Since , is -small in and , and then, is a -lifting module by Theorem 38.
Now, we investigate when the factor module of a -lifting module is -lifting.

Proposition 10. Let be a -lifting module. For any , the module is -lifting if one of the following statements is provided:(1)For any , .(2) is a distributive module.(3) for any idempotent . Most particularly, is fully invariant.

Proof. (1)Let . Since and is -lifting, there exists with and . It is clear to verify that and . Since , then by Lemma 3. Hence, is -lifting.(2)This condition will be proved by using (1). Let . We have , and by hypothesis, . Hence, and so is -lifting.(3)Let . By (1), we will show that . Let be the projection map where . Then, and . By assumption, and . So we have and . Therefore, . From here, it is clear to see that and . This implies . In addition, since , we have . Hence, is -lifting by (1).In Lemma 15, we proved that each direct summand of a -lifting module is -lifting. But the contrast idea is not true generally. By Theorem 12, we present a way verifying this claim by adding suitable conditions. But first, we give the following useful lemma see ([6], 41.14) for completeness.

Lemma 11. Let . Then, the following conditions listed are equivalent:(1) is -projective(2)For each with , there exists providing

Theorem 12. Let be a module such that is both -projective and -projective. If and are -lifting modules, then so is .

Proof. Let . In that case, as is -lifting , there exist direct summands of with and . So we have . Since is self and -projective it is clear to say that is -projective. By taking into account the exact sequence , it can be seen that is -projective [[6], 18.1/18.2]. Therefore, by Lemma 18, there exists providing . Following this, we can say for any . In addition, since is -lifting, there exists such that and for any . Therefore, the fact that can be seen easily. Since and , we have and so . In addition, .
Recall that the family of relatively projective modules is defined as a family of modules where is -projective for each distinct .

Corollary 13. Let be a semisimple module and be a -lifting module which are relatively projective with , then is -lifting.

In the next proposition, we verify that the direct sum of two -lifting modules is -lifting for a duo module (whose submodules are all fully invariant).

Proposition 14. Let be a duo module. If and are -lifting modules, then so is .

Proof. Let . Since is a duo module, it can be written that . By assumption, for the submodules and , there exist submodules , and , , respectively, such that , , and and , , and . Therefore, . So we have and .
In the following example, a type of a module can be seen that is -lifting but not -lifting.

Example 2. Let where be a field and . Let , where is the matrix unit in . Note that for the idempotent ideal , we have a hereditary torsion theory with the torsion part -. Let . Note that is simple, and it is not a direct summand of as which is a direct summand. Also, is not -torsion as . Thus, does not involve any direct summand of such that is -torsion. Hence, is not -lifting. However, is a lifting and so a -lifting module by [14].

4. -Supplemented Modules

In this part of the study, we define -supplemented modules and present basic properties of this type of modules.

Lemma 15. Let ,. Then, the statements given below are equivalent:(1) and (2), for any proper with being -singular,

Proof. If , where and is -singular, then . Hence, we have since .
If , where and is -singular, then . By (2), . So .

Definition 16. is said to be a -supplement submodule of in if and provide one of the equivalent conditions given in Lemma 19. By the way, is called -supplemented if each submodule of has a -supplement in .
We cannot claim every -supplemented module is -supplemented or the converse statement directly because of being specialized versions of each other.
It can be seen that in the following proposition, being -supplemented is preserved by homomorphic images.

Proposition 17. Every homomorphic image of a -supplemented module is -supplemented.

Proof. Let be a -supplemented module, be an epimorphism and be a submodule of . By assumption, there exists providing and . In that case, and by Lemma 5. Thus, is a -supplement of in . Hence, is -supplemented.

Lemma 18. Let be a module and , , . If is a -supplement of in and is a -supplement of in , then is a -supplement of in .

Proof. Because is a -supplement of in , we get , , and is a -supplement of in ; we have , . It is enough to show that . Let with and -singular. Then, . Since , for a non--singular submodule with by Lemma 3. Hence, by the modular law. Since is -singular and , we have . Thus, as is both -singular and non--singular. Finally, is obtained.

Lemma 19. For a -supplemented module , is a semisimple module.

Proof. Let . There exists providing and . So . Thus, and . So we have .
Clearly, each -lifting module is -supplemented. The converse might be provided under additional conditions as in the following proposition.

Proposition 20. A projective -supplemented module is -lifting whenever each -supplement submodule of is a direct summand.

Proof. Let . By hypothesis, there exists with and . Since , for some . Following that, we have as is projective and . Hence, we obtain a decomposition of such that with and ; that is, is -lifting.
Let be an -module and and be -torsion free modules. If is -supplemented, then is also -supplemented and vice versa.
Let be a stable torsion theory (as Goldie torsion theory) or is -torsion free (as . Then, every -supplemented module is also -supplemented.
Let be -torsion and nonsingular ring. Then, a left -module is -supplemented module if and only if is -supplemented.
Before giving the finite sum of -supplemented modules which is also -supplemented, we need the following lemma.

Lemma 21. Let , , and be a -supplemented module. If has a -supplement in , then so does .

Proof. Since has a -supplement in , there exists providing and . Also, there exists providing and . Thus, we have and ; that is, is a -supplement of in . Now, we claim that is a -supplement of in . It is evident that and since and .

Proposition 22. Let and be -supplemented modules. If , then is a -supplemented module.

Proof. Let . Since has a trivial -supplement 0 in and is -supplemented, has a -supplement in by Lemma 21. Thus, has a -supplement in as is -supplemented by Lemma 21. So, is -supplemented.
Recall that for a module a module is called finitely -generated if there exists an epimorphism from the sum of finitely many copies of to .
As an immediate consequence of the finite sum and homomorphic image property, we give the following proposition.

Proposition 23. If is a -supplemented module, then every finitely -generated module is -supplemented.

Proof. Let be a -supplemented module and be a finitely -generated module. Then, there exists an epimorphism from ( is a finite index set) to . Since is -supplemented, then is a -supplemented module by Propositions 17 and 22.

5. Amply -Supplemented Module

In this part of the study, we define amply -supplemented modules and give basic properties of them. Also, we present relations between these modules and the modules introduced in previous sections.

Definition 24. A module is called amply-supplemented if for any submodules , of with , there exists a -supplement of in contained in .
Clearly, every -lifting module is amply -supplemented, and every amply -supplemented module is -supplemented.

Proposition 25. Every homomorphic image of an amply -supplemented module is amply -supplemented.

Proof. Let be an amply -supplemented module and be a homomorphism from to . We claim that is amply -supplemented. Let . Then, . Thus, there exists a submodule of contained in with , . Following that, we have and .

Proposition 26. Let be a module. If every submodule of is -supplemented, then is an amply -supplemented module.

Proof. Let for , . By hypothesis, there exists providing and . Thus, we have . Hence, is an amply -supplemented module.

Corollary 27. The following listed statements given are equivalent for a ring :(1)Every -module is amply -supplemented(2)Every -module is -supplementedRecall that we say a module is -projective if there exists a homomorphism such that and for every submodule , which satisfies .

In general, every projective module is -projective [1].

Theorem 28. Let be a -projective -supplemented module, then is an amply -supplemented module.

Proof. For any submodule of , let for . As is -projective, there exists an endomorphism of providing and . Let be a -supplement of in . Then, , so we have with . Also, . Hence, is a -supplement of in contained in .

Corollary 29. If is a projective and -supplemented module, then is an amply -supplemented.

Theorem 30. Let be an amply -supplemented module whose -supplements are direct summands of . Then, is a -lifting module.

Proof. Since is amply -supplemented, there exists a -supplement for every and there exists a -supplement for with , . Then, we have , and is obtained. For the projection map , it is true that . Moreover, as , and so . Hence, for every , there exists a decomposition of providing with and .

Corollary 31. Let be a projective -supplemented module whose -supplements are direct summands of . Then, is a -lifting module.

Proof. It is clear from Theorem 30 and Corollary 31.

6. -Semiperfect Modules

In this section, first, we define the projective -cover of a module by means of -small submodules to get the concept of -semiperfect modules. At the end, we give a characterization theorem between -semiperfect modules and (amply) -supplemented modules.

Definition 32. Let be a (projective) module and be an epimorphism with . In this case, is called a (projective) -cover of .

Definition 33. A module is called a -semiperfect module if any homomorphic image of has a projective -cover.

Proposition 34. If is an epimorphism with , then .

Proof. It is clear from [[15], Corollary, 8.17].

Lemma 35. Let and be -covers, then is a -cover.

Proof. Since and are -covers, then and . Now, we claim that . Let with -singular. Following that, we have . Hence, is obtained as , and is -singular. This implies that . Therefore, we have since -singular and .

Lemma 36. Let be -covers for every . Then, is a -cover.

Proof. It can be proved by the standard way.

Theorem 37. Let be a module and . Then, the following listed statements are equivalent:(1) has a projective -cover.(2)If for , then has a -supplement such that has a projective -cover.(3) has a -supplement which has a projective -cover.

Proof. Let be a projective -cover. Since , is an epimorphism. Since is projective, there exists a homomorphism from to satisfying . Following that, we have and so , . Also, , since . Hence, is a -supplement of in . Thus, is a projective -cover as .
It is clear.
Let be a projective -cover. By hypothesis, and . It follows that the natural epimorphism is a -cover. So is a projective -cover.

Theorem 38. The following listed statements are equivalent for a module :(1) is -semiperfect(2) is amply -supplemented whose -supplements have projective -covers(3) is -supplemented whose -supplements have projective -covers

Proof. It is evident by Theorem 37.

Data Availability

No underlying data were collected or produced in this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest.

Authors’ Contributions

E. O. Sozen conceptualized the study and proposed the methodology. E. O. Sozen and A.R. Moniri Hamzekolaee validated the study. E. O. Sozen carried out the investigation and gathered the resources. E. O. Sozen wrote the original draft. E. O. Sozen, A. R. Moniri Hamzekolaee, and J.Tian wrote the review and edited the manuscript. J. Tian acquired funding.

Acknowledgments

This work was supported by the Software Engineering Institute of Guangzhou, 510990 of P. R. China, under the Grant/Award number (ST202101).