#### 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.