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# -Regular Modules

**Academic Editor:**Jong Hae Kim

#### Abstract

We introduced and studied -regular modules as a generalization of -regular rings to modules as well as regular modules (in the sense of Fieldhouse). An -module is called -regular if for each and , there exist and a positive integer such that . The notion of -pure submodules was introduced to generalize pure submodules and proved that an -module is -regular if and only if every submodule of is -pure iff is a -regular -module for each maximal ideal of . Many characterizations and properties of -regular modules were given. An -module is -regular iff is a -regular ring for each iff is a -regular ring for finitely generated module . If is a -regular module, then .

#### 1. Introduction

Throughout this paper, unless otherwise stated, is a commutative ring with nonzero identity and all modules are left unitary. For an -module , the annihilator of in is . The symbol □ stands for the end of the proof if the proof is given or the end of the statement when the proof is not given.

Recall that a ring is said to be regular (in the sense of von Neumann) if for each , there exists such that [1]. The concept of regular rings was extended firstly to -regular rings by McCoy [2], recall that a ring is -regular if for each , there exist and a positive integer such that [2] and secondly to modules in several nonequivalent ways considered by Fieldhouse [3], Ware [4], Zelmanowitz [5], and Ramamurthi and Rangaswamy [6]. In [7], Jayaraman and Vanaja have studied generalizations of regular modules (in the sense of Zelmanowitz) by Ramamurthi [8] and Mabuchi [9]. Following [10], we denoted Fieldhouse’ regular modules by -regular. An -module is called -regular if each submodule of is pure [3].

Dissimilar to the generalizations that have been studied in [7, 9] and [8], in this paper a new generalization of -regular rings to modules and -regular modules was introduced, called -reular (generalized -regular) modules. An -module is called -regular if for each and , there exist and a positive integer such that . A ring is called -regular if is -regular as an -module. On the other hand, -regular modules are also a generalization of -regular rings. Thus, is a -regular ring if and only if is a -regular -module. Furthermore, we introduced a new class of submodules, named, -pure submodules as a generalization of pure submodules. A submodule of an -module is said to be -pure if for each , there exists a positive integer such that . Recall that a submodule of an -module is pure if for each ideal of [11]. We find that the relationship between -regular modules and -pure submodules is an analogous relationship between -regular modules and pure submodules.

In Section 3.1 of this paper, after the concept of -regular modules was introduced, we obtained several characteristic properties of -regular modules. For instance, it was proved that the following are equivalent for an -module : (1) is -regular; (2) every submodule of is -pure; (3) is a -regular ring for each ; (4) and for each and , there exist and a positive integer such that . It is also shown that if is a finitely generated -module, then is -regular if and only if is a -regular ring.

Section 3.2 was devoted to investigate the relationship between -regular modules with the localization property and semisimple modules. For example, we proved that is a -regular -module if and only if is a -regular -module for every maximal ideal of if and only if is a semisimple -module for every maximal ideal of .

Finally, in Section 3.3 we studied some properties of the Jacobson radical, , of -regular modules. Thus we proved that if is a -regular -module, then , and also we get that if is a reduced ideal of a ring and is a -regular -module, then .

#### 2. The Notion of -Regular Modules and General Results

We start by recalling that an -module is -regular if each submodule of is pure [3], and a ring is -regular if for each , there exist and a positive integer such that [2].

*Definition 1. *An -module is called -regular if for each and , there exist and a positive integer such that . A ring is -regular if and only if is -regular as an -module.

The following gives another characterization for -regular modules.

Proposition 2. *An -module is -regular if and only if is a -regular ring for each .*

*Proof. *Suppose that is a -regular -module, so for each and , there exist and a positive integer such that ; hence, which means that ; therefore, is a -regular ring. Conversely, suppose that is a -regular ring for each , thus for each , there exist and a positive integer such that ; hence, which implies that ; therefore, is a -regular -module.

It is clear that every -regular module is -regular, but the converse may not be true in general; for example, by applying Proposition 2 to the -module , we can easily see that it is -regular; however, is not an -regular -module. In fact, the -module is -regular for each positive integer [12], while it is not -regular for some positive integer . On the other hand, the -module is not -regular because for each we have that , but which is not a -regular ring [12].

*Remark 3. *(1)If is a -regular ring, then every -module is -regular.(2)Every module over Artinian ring is -regular (because every Artinian ring is -regular [12]).(3)A ring is -regular if and only if is -regular as an -module.(4)Every submodule of a -regular module is -regular module. In particular, every ideal of a -regular ring is -regular -module. Furthermore, it follows from (1) that if is an ideal of a -regular ring , then the -module is -regular.(5)The converse of (1) is true if the module is free, that is, any free -module is -regular if and only if is a -regular ring. For if, is a free -module, then for each , so is a -regular ring.(6)If an -module is -regular and it contains a nontorsion element, then is a -regular ring. In particular, if is a -regular -module and is not a -regular ring, then is a torsion -module.

Now from Proposition 2 and Remark 3(3), we conclude the following.

Corollary 4. *The following statements are equivalent for a ring:*(1)* is a -regular ring;*(2)* is a -regular ring for each . *

We have seen previously that every -regular -module is -regular. In the following we consider some conditions such that the converse is true.

*Remark 5. *(1)Let be a reduced ring. An -module is -regular if and only if is a -regular -module.(2)An -module is -regular if and only if is a -regular -module and for each, where is the prime radical of the ring .

Now, we describe -regular modules over the ring of integers .

Proposition 6. *A -module is -regular if and only if is a torsion -module.*

*Proof. *If is a -regular -module, then by Remark 3(6) is a torsion -module. Conversely, if is a torsion -module, then for some positive integer ; hence, is a -regular ring for each positive integer [12], which implies that is a -regular -module.

Proposition 7. *Every homomorphic image of a -regular -module is -regular. *

* Proof. *Let , be two -modules such that is -regular and let be an -epimorphism. For every , there exists such that . It is clear that . Define by for each . It is an easy matter to check that is well defined -epimorphism. Since is a -regular ring, then is also a -regular ring [12]. Therefore, is a -regular -module.

Corollary 8. *The following statements are equivalent for an -module :*(1)* is a -regular -module for every nonzero submodule of .*(2)* is a -regular -module for every . *

Another characterization of a -regular -module is given in the next result.

Proposition 9. *An -module is -regular if and only if for each and , there exist and a positive integer such that .*

*Proof. *Suppose that is a -regular -module, so for each and , there exist and a positive integer such that , then we can take and hence . Conversely, for each and , there exist and a positive integer such that . Now, (after times), thus where which implies that is a -regular -module.

#### 3. Main Results

##### 3.1. -Regular Modules and Purity

Recall that a submodule of an -module is pure in if each finite system of equations which is solvable in , is solvable in [13]. It is not difficult to prove that is pure in if and only if for each ideal of , [11]. This motivates us to introduce the following definition as a generalization of pure submodules.

*Definition 10. *A submodule of an -module is called -pure if for each , there exists a positive integer such that .

It is clear that every pure module is -pure.

The following theorem gives another characterization of -regular modules in terms of -pure submodules.

Theorem 11. *An -module is -regular if and only if every submodule of is -pure.*

*Proof. *Suppose that is a -regular -module and let be any submodule of . For each and for some positive integer , let , then there exists such that . Since is -regular, then there exists such that . Put , then which implies that , but , so and hence . On the other hand, it is clear that , thus which means that is a -pure submodule.

Conversely, assume that every submodule is -pure and let and such that which is a -pure submodule of for some positive integer , then for each . In particular, if we get which implies that there exists such that , so is a -regular -module.

Corollary 12. *An -module is -regular if and only if for each , there exist and a positive integer such that is a -pure submodule. *

*Remark 13. *Fieldhouse in [11] proved that for a submodule of an -module , if is a flat -module, then is pure. On the other hand, if is flat and is pure, then is flat. So, immediately we have that for a flat -module, if is a flat -module for each submodule of , then is -regular -module. It is not difficult to prove that in case of -regular modules the converse of the latest statement is true; however, we do not know whether it is true for -regular modules or not.

*Remark 14. *In [14], Mao proved that a right -module is -flat if and only if there exists an exact sequence with free such that for any , there exists a positive integer satisfying , where (1) a right -module is said to be generalized -flat (-flat for short) if for any , there exists a positive integer (depending on ) such that the sequence is exact [15], (2) a right -module is -flat [16] or torsion-free [15] if for any , the sequence is exact. Obviously, every flat module is -flat [16] and every -flat module is -flat [14].

According to the above remark we get the following.

Corollary 15. *An -module is -flat if and only if there exists an exact sequence with is a submodule of a free -module such that is a -pure submodule. *

Corollary 16. *For every submodule of a free -module , if there exists an exact sequence such that is a -pure submodule in , then is a -flat -module if and only if is -regular.*

Now, we recall that (1) an -module is -injective if for every principal ideal of , every -homomorphism of into extends to one of into [17]. A ring is called -injective if is -injective as an -module. (2) An -module is called -injective if for any , there exists a positive integer such that and any -homomorphism of into extends to one of into . A ring is called -injective if is -injective as an -module [18]. -injective modules are called -injective modules by some other authors [19–22]. (3) An -module is called -injective (weak -injective) if for any , there exists a positive integer such that every -homomorphism of into extends to one of into ( may be zero). A ring is called -injective if is -injective as an -module [23–25]. (4) A ring is called . if every principal ideal of is projective. And is called -ring if for any , there exists a positive integer (depending on ) such that is projective [26, 27].

Note that -injectivity implies -injectivity (or -injectivity) and -injectivity, as well as the concept of . rings implies the concept of -rings. However, the notion of -injective (or -injective) modules is not the same notion of -injective modules.

It is known that a ring is -regular if and only if every -module is -injective [12, 22], so from all the above we conclude the following theorem.

Theorem 17. *The following statements are equivalent for a ring .*(1)* is a -regular ring.*(2)* is a -regular ring for each .*(3)*Any free -module is -regular.*(4)*Every -module is -injective.*

We end this section by the following two related results.

Proposition 18. *Let be an -module. If is a -regular ring, then is a -regular -module.*

*Proof. *We have that for each , so there exists an obvious -epimorphism defined by . Since is a -regular ring, then is a -regular ring [12]; therefore, is a -regular -module.

In case of finitely generated modules, the converse of Proposition 18 is true.

Proposition 19. *Let be an -module. If is a finitely generated -regular -module, then is a -regular ring.*

*Proof. *Let be a finite set of generators of . Put , and , , then . Now define by = for each . It is easily checked that is a ring monomorphism. Thus, can be identified with a subring of . In fact

We will show now that , and hence is a -regular ring. Since is a -regular -module, then is a -regular ring, thus for each and , there exist and a positive integer such that ; this means that . Define by the relation , then which implies that for each , , so is a -regular ring and hence is a -regular ring.

##### 3.2. -Regular Modules and Localization

In this section we study the localization property and semisimple modules with -regular modules and we give some characterizations of -regular modules in the sense of them.

Theorem 20. *Let be an -module. is a -regular -module if and only if is a -regular -module for each maximal ideal in .*

*Proof. *Let be a -regular -module, and let be any maximal ideal in . Let and , where , and , . So there exist and a positive integer such that . Hence, = , where , then is -regular -module.

Conversely, suppose that is a -regular -module. Let be a submodule of and let be a maximal ideal of . By Theorem 11, is a -pure submodule of ; therefore, for each and for some positive integer . But by [28], we have that and , then , again by [28], we get that , which implies that is a -pure submodule of and by Theorem 11 , is a -regular -module.

Recall that an -module is simple if 0 and are the only submodules of , and an -module is said to be semisimple if is a sum of simple modules (may be infinite). A ring is semisimple if it is semisimple as an -module [29]. It is known that over any ring , a semisimple module is -regular [4, 30], consequently it is -regular. Furthermore, it is known that over a local ring, every -regular module is semisimple [31]. We can generalize the latest statement as the following.

Proposition 21. *Every -regular module over local ring is semisimple.*

*Proof. *Let be the only maximal ideal of . Since is -regular, then for each we have that is -regular local ring which implies that is a field [12]; hence, is a maximal ideal, so for each . Therefore, . On the other hand, is a field, which implies that is a vector space over the field which is a simple ring. Then is a semisimple module over the ring . Thus, is a semismple -module [29].

As an immediate result from Theorem 20 and Proposition 21, we get the following.

Corollary 22. *Let be an -module. is -regular if and only if is a semisimple -module for each maximal ideal of .*

We mentioned before that every -regular -module is -regular; the following gives us another condition such that the converse is true.

Corollary 23. *Let be a local ring. An -module is -regular if and only if is a -regular -module.*

Corollary 24. *An -module is -regular if and only if and are -regular -modules.*

*Proof. *Assume that and are -regular -modules, then for each maximal ideal in , each of and is a semisimple module (Proposition 21); hence, it is an easy matter to check that is a semisimple module, so is a -regular module. Thus, is a -regular module (Theorem 20). The other direction is obtained directly from Proposition 7.

Finally we can summarize that the conditions under which -regular modules coincide with -regular modules and the characterizations of -regular modules, of Section 2 with those of this section, in the following Proposition 25 and Theorem 26, respectively:

Proposition 25. *An -module is -regular if and only if is an -regular module, if any of the following conditions are satisfied.*(1)* is a local ring.*(2)* is a reduced ring.*(3)*The prime radical of the ring ** is zero for each *.

Theorem 26. *The following statements are equivalent for a ring .*(1)* is a -regular -module.*(2)* is a -regular ring for each *(3)*For each and , there exist and positive integer such that .*(4)*Every submodule of is -pure.*(5)*For each , there exist and a positive integer such that is a -pure submodule.*(6)* is a -flat -module, if for every submodule of a free -module there exists an exact sequence such that is a -pure submodule in .*(7)*If is a finitely generated -module, then is a -regular ring.*(8)* is a -regular -module for each maximal ideal in .*(9)* is a semisimple -module for each maximal ideal of .*

##### 3.3. The Jacobson Radical of -Regular Modules

Let be an -module. A submodule of is said to be small in if for each submodule of such that , we have [32]. The Jacobson radical of a ring will be denoted by . The following submodules of are equal: (1) the intersection of all maximal submodules of , (2) the sum of all the small submodules of , and (3) the sum of all cyclic small submodules of . This submodule is called the Jacobson radical of and will be denoted by [29, 32].

It is appropriate now to note that for each element it may happen that . But some cases demand that must be nonzero element. For this purpose we introduce the following concept.

*Definition 27. *An -module is called -regular if for each and , there exist and a positive integer with such that . A ring is called -regular if it is -regular as an -module.

It is clear that -regularity implies -regularity and they are coincide if is a reduced ring.

Proposition 28. *Let be an -regular -module, then .*

*Proof. *For each and for each , there exist and a positive integer with such that , then . If , then and is invertible, so , but we have that and ; hence, which implies that .

Recall that an -module is faithful if for every such that implies [29], or equivalently, an -module is called faithful if [33].

Corollary 29. *If is a faithful -regular -module, then .*

Corollary 30. *Let be a reduced ring and be a -regular -module, then .*

Corollary 31. *Let be any ring such that is a reduced ideal of and let be a -regular -module, then .*

Corollary 32. *Let be a reduced ring. If is a faithful -regular -module, then .*

It is suitable to mention that, in general, not every module contains a maximal submodule; for example, as -module has no maximal submodule. So we have the next two results, but first we need Lemma 33 which is proved in [29].

Lemma 33. *An -module is semisimple if and only if each submodule of is direct summand.*

Proposition 34. *Let be a -regular -module, then .*

*Proof. *Since is a -regular -module, then is a semisimple -module for each maximal ideal of (Corollary 22). Since each cyclic submodule of is direct summand (Lemma 33), then it cannot be small; therefore, the Jacobson radical of a semisimple module is zero, so for each maximal ideal of . On the other hand, [28], thus for each maximal ideal of , and hence [28].

Corollary 35. *Every nonzero -regular -module contains a maximal submodule.*

*Proof. *Suppose not, then , but (Proposition 34), so which is a contradiction.

Corollary 36. *Let be a -regular -module, then for each , there exist a maximal submodule such that .*

*Proof. *If , for each maximal submodule of , then which implies that .

Corollary 37. *Let be a -regular -module, then every proper submodule of contained in a maximal submodule.*

*Proof. *Let be a proper submodule of . Since is a -regular -module, then is -regular (Proposition 7), so contains a maximal submodule (Corollary 35), which means that there exists a submodule of such that , is a maximal submodule of ; therefore, is a maximal submodule of and contains .

Corollary 38. *Every simple submodule of a -regular -module is direct summand.*

*Proof. *Let be a simple submodule of a -regular -module , then is cyclic; say , then there exists a maximal submodule of such that (Corollary 37). It is clear that . Now, if , then because is a simple submodule. Thus, which is a contradiction, so .

#### Acknowledgment

S. Chen was sponsored by Project no. 11001064 supported by the National Natural Science Foundation of China.

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#### Copyright

Copyright © 2013 Areej M. Abduldaim and Sheng Chen. 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.