#### Abstract

In this paper, we investigate the notions of -projective, -injective, and -flat modules and give some characterizations of these modules, where is a class of left modules. We prove that the class of all -projective modules is Kaplansky. Further, if the class of all -injective -modules is contained in the class of all pure projective modules, we show the existence of -projective covers and -injective envelopes over a -hereditary ring. Further, we show that a ring is Noetherian if and only if -injective -modules coincide with the injective -modules. Finally, we prove that if every module has a -injective precover over a coherent ring, where is the class of all pure projective -modules and is the class of all -modules.

#### 1. Introduction

Throughout this paper, denotes an associative ring with identity and all -modules, if not specified otherwise, are left -modules. - denotes the category of left -modules.

The notion of -injective modules over arbitrary rings was first introduced by StenstrĂ¶m in [1]. An -module is called -injective (absolutely pure) if for all finitely presented -modules . Let be a class of left -modules. Mao and Ding in [2] introduced the concept of -injective modules (see Definition 6). Selvaraj et al. developed -injective and -flat -modules and studied covers and envelopes of modules with Goresntein properties in [3â€“5].

A pair is a cotorsion theory (see Definition 4). In our article, is a cotorsion theory generated by the class [6], that is, and If is a hereditary ring (that is, every ideal is projective), then is closed under submodules and containing all injective -modules in the cotorsion theory . In this case, results are trivial. For this reason, we introduced -hereditary ring (see Definition 16), and we restrict the setting to -hereditary rings.

The notions of (pre)covers and (pre)envelopes of modules were introduced by Enochs in [7] and, independently, by Auslander and SmalĂ¸ in [8]. Since then, the existence and the properties of (pre)covers and (pre)envelopes relative to certain submodule categories have been studied widely. The theory of (pre)covers and (pre)envelopes, which play an important role in homological algebra and representation theory of algebras, is now one of the main research topics in relative homological algebra.

Salce introduced the notion of a cotorsion theory in [9]. Enochs showed the important fact that closed and complete cotorsion pairs provide minimal versions of covers and envelopes. Eklof and Trlifaj [10] proved that a cotorsion pair is complete when it is cogenerated by a set. Consequently, many classical cotorsion pairs are complete. In this way, Bican et al. [11] showed that every module has a flat cover over an arbitrary ring. These results motivate us to define -projective -modules (see Definition 14) and -pure projective -modules (see Definition 21), and we prove the existence of -projective cover and -injective envelope. In particular, we denote by the class of all pure injective -modules, and we prove the following main result.

Theorem 1. *Let be a -hereditary ring and . Then, every -module has a -projective cover and an -injective envelope.*

Self-injective rings were introduced by Johnson and Wong in [12]. A ring is said to be self injective if over itself is an injective -module. In this paper, we introduce self--injective ring (see Definition 27). Recall that an -module is called *reduced* [13] if it has no nonzero injective submodules. An -module is said to be *coreduced* [14] if it has no nonzero projective quotient modules. Mao and Ding [15] proved that an -injective -module decomposes into an injective and a reduced -injective -module over a coherent ring. Similarly, we prove the following result.

Theorem 2. *Let be a self -injective and -hereditary ring and . Then, an -module is -projective if and only if is a direct sum of a projective -module and a coreduced -projective -module.*

Pinzon [16] proved that every module has an -injective cover over a coherent ring. We prove the following result that provides the existence of -injective cover, where is the class of all pure projective -modules and the class of all --modules (see Definition 41).

Theorem 3. *Let be a coherent ring and Then, every -module has a -injective cover.*

This paper is organized as follows: In Section 2, we recall some notions that are necessary for our proofs of the main results of this paper.

In Section 3, we investigate the notions of -injective and -flat modules and give some characterizations of these classes of modules.

In Section 4, we introduce -hereditary ring. Further, we investigate -projective module and give some characterizations. Further, we prove Theorems 1 and 2.

In Section 5, we prove that if is a submodule of an -injective -module , then, is a special -injective envelope of if and only if is an -projective essential extension of

In the last section, we assume that is the class of all pure projective -modules and we prove that every -module has a -injective preenvelope. Finally, we prove the main theorem of this section Theorem 3.

#### 2. Preliminaries

In this section, we recall some known definitions and some terminology that will be used in the rest of the paper.

Given a class of left -modules, we write

Following [7], we say that a map with is a -precover of , if the group homomorphism is surjective for each . A -precover of is called a -cover of if is right minimal, that is, if implies that is an automorphism for each . is a precovering class (resp., covering class) provided that each module has a -precover (resp., -cover). Dually, we have the definition of -preenvelope (resp., -envelope).

A -precover of is said to be *special* [6] if is an epimorphism and .

A -preenvelope of is said to be *special* [6] if is a monomorphism and .

A -envelope is said to have the *unique mapping property* [17] if for any homomorphism with , there is a unique homomorphism such that

A module is said to be *pure projective* [18] if it is projective with respect to pure exact sequence.

An -module is called super finitely presented ([19]) if there exist a projective resolution with each projective -module is finitely generated.

Following [20, 21], an -module is called *weak injective* if for every super finitely presented -module . A right -module is called *weak flat* if for every super finitely presented -module .

A class of left -modules is said to be *injectively resolving* [6] if contains all injective modules and if given an exact sequence of left -modules

whenever

*Definition 4. *A pair of classes of modules is called a *cotorsion theory* [6] if and (1)A cotorsion theory is said to be *perfect* [22] if every module has an -cover and a -envelope

*Definition 5. *A cotorsion theory is said to have enough injectives [13] if for every -module there is an exact sequence with and

For an -module , denotes the flat dimension of and denotes the injective dimension of

The -coresolution dimension of , denoted by , is defined to be the smallest nonnegative integer such that for all -modules (if no such exists, set ), and is defined as

We denote by the ring of all integers and by the field of all rational numbers. For a left -module , we denote by the *character module* of denotes the class of all injective left -modules.

For unexplained terminology, we refer to [23, 24].

#### 3. -Injective and -Flat Modules

In this section, we assume that is any class in We begin with the following definition:

*Definition 6 (see [2]). *A left -module is called -injective if for all left -modules A right -module is said to be -flat if for all left -modules

We denote by the class of all -injective modules and the class of all -projective -modules and, further, the class of all -flat -modules.

*Example 7. *(1)If then, is the class of all injective -modules and is the class of all flat right -modules.(2)If is the class of all finitely presented -modules, then, is the class of all -injective -modules.(3)If is the class of all left super finitely presented -module, then, is the class of all weak injective -modules and is the class of all weak flat right -modules.

Proposition 8. *Let be an -module. Then, the following are true:
*(1)*Let Then, is injective if and only if is -injective and *(2)* is injective if and only if is -injective and *

*Proof. *(1)The direct implication is clear. Conversely, let be -injective and . For any -module , consider an exact sequence with an injective envelope of . We have an exact sequenceSince , , and hence, is injective.
(2)The direct implication is clear by the definition of an injective module. Conversely, let be -injective and . Consider an exact sequence with an injective envelope of For any -module we have an exact sequence . Since , , and hence, is -injective. Therefore, , so that the exact sequence is split. It follows that is a direct summand of as desired.

We now give some of the characterizations of -injective module:

Proposition 9. *Let The following are equivalent for a left -module *(1)* is -injective*(2)*For every exact sequence , with is an -precover of *(3)* is a kernel of an -precover with an injective module*(4)* is injective with respect to every exact sequence , with *

*Proof. *. Consider an exact sequence
where Then, by hypothesis, is surjective for all left -modules as desired.

. Let be an injective hull of and consider the exact sequence Since is injective, it belongs to So assertions holds.

. Let be a kernel of an -precover with an injective module. Then, we have an exact sequence . Therefore, for any left -module the sequence is exact. By hypothesis, is surjective. Thus, , and hence (1) follows

. Consider an exact sequence
where Then, is surjective, as desired.

. For each left -module there exists a short exact sequence with a projective -module, which induces an exact sequence . By hypothesis, is exact. Thus, , and hence (1) follows.

The following note is useful for understanding the notations in Examples 11 and 15.

*Note 10. *From Introduction, We get a new cotorsion theory generated by the class

*Example 11. *Let be a commutative Noetherian and complete local domain. Assume that the and . Then, is an -injective -module, where is the class of all -injective -modules.

*Proof. *Consider the residue field and an exact sequence . If is an -module, the sequence is exact. By ([25], p 43), is an injective cover of Since then, the class of all injective -modules and the class of all -injective -modules are equal, that is, . Clearly, is an -injective cover of . Thus, is surjective for every -injective -module . We get for every -injective -module , and hence, is -injective. On the other hand, is injective and so -injective. Therefore, is -injective.

We now give some characterizations of -flat module:

Proposition 12. *The following are equivalent for a right -module :
*(1)* is -flat*(2)*For every exact sequence with the functor preserves the exactness*(3)* for all *(4)* is -injective*

*Proof. *. Consider an exact sequence with Since is -flat, . Hence, the functor preserves the exactness.

. Let Then, there exists a short sequence with a projective module, which induces an exact sequence . By hypothesis, . Thus, is -flat.

It follows from the natural isomorphism ([26], p 34) .

It follows from the natural isomorphism ([27], VI 5.1) .

Proposition 13. *Let be a coherent ring. Then, a right -module is flat if and only if is -flat and .*

*Proof. *The â€śonly ifâ€ť part is trivial. Conversely, suppose that is -flat. By Proposition 12, is -injective. By ([28], Theorem 2.1), . Then since . By Proposition 8, is injective and hence is flat.

#### 4. -Projective Cover and -Injective Envelope

Now, we introduce -projective modules.

*Definition 14. *An -module is called -projective if for all -modules .

*Example 15. *Let be a complete local ring and be the class of all -injective -modules. Assume that the . Then is -projective, where is the residue field and is an -injective envelope of .

*Proof. *By Note 10, we get a cotorsion theory generated by the class Consider an exact sequence where is an injective envelope of Since then, the class of all injective -modules and the class of all -injective -modules coincide, that is, . It follows that is an -injective envelope of Since the class of all -projective modules is closed under extensions, then by Lemma 2.1.2 in [25], is -projective.

We now introduce Definition 16.

*Definition 16. *A ring is called left -hereditary if every left ideal of is -projective.

*Remark 17. *If is the class of injective left -modules, then, every ring is -hereditary. It is also easy to see that a ring is left hereditary if and only if is -hereditary for every class of left -modules.

Given a class of left -modules, we denote by the class

*Example 18. *Let be a commutative Noetherian ring. If then is a -hereditary ring.

*Proof. *Let be an ideal of . We claim that is -projective. By hypothesis, for all and for all It follows that Thus, for all ideals of and for all Consider an exact sequence So for all Hence, is -projective, as desired.

Note that an -module is -projective if for all -modules . Clearly, is a cotorsion theory.

Proposition 19. *A ring is left -hereditary if and only if every submodule of a -projective left -module is -projective.*

*Proof. *Let be left -hereditary and be a left ideal of . Then, there is an exact sequence By hypothesis, for any -injective left -module Thus, Let be a submodule of an -projective left -module Then, for any -injective left -module , the sequence
is exact. Thus, since is -projective and The reverse implication is clear.

In general, is not closed under pure submodules; for example, if is the class of flat modules, then and this class is not closed under pure submodules in general. As an easy consequence of Proposition 19, we have that the class is closed under pure submodules over a -hereditary ring.

*Definition 20 (see [22]). *Let be a class of -modules. Then, is said to be Kaplansky class if there exists a cardinal such that for every and for each , there exists a submodule of such that and and

*Definition 21. *An -module is called -pure injective if extends to a homomorphism for all where (that is, is a pure submodule of ). We denote by the class of all -pure projective modules.

*Remark 22. *The class of all pure injective -modules is contained in the class of all -pure injective -modules.

*Example 23. *(1)Clearly, is a cotorsion theory. Then, the class of all -pure injective -modules is the class of all pure injective -modules.(2)If is the class of all flat -modules and is the class of all cotorsion -modules, then, is a cotorsion theory. Then, the class of all -pure injective -modules is the class of all cotorsion -modules.If is Noetherian and then, the class of all -pure injective -modules is

*Proof. *(1)Straightforward(2)Let be a -pure injective -module. We show that is a cotorsion -module. For any flat -module , there is a pure exact sequence where is a projective -module. Then, we get the following exact sequence:It follows that because is surjective. Hence, is cotorsion. Conversely, let be a cotorsion -module. Suppose the following sequence:
where is pure exact. Then, we get the exact sequence It follows that since is flat. Hence , is -pure injective.
(3)By Exercises 6 in [13], is a cotorsion theory. Let be an -pure injective -module. We show that For any -module consider the following exact sequence:where is projective. By hypothesis, is injective. Since the cotorsion theory has enough injectives, is injective, that is It follows that the above sequence is pure exact. Then, we get the exact sequence Thus, since is -pure injective. Hence, Conversely, Let be an -injective -module. Suppose the following exact sequence:
where is pure exact. It follows that since the cotrosion theory has enough injectives. Then, is surjective. Hence, is -pure injective.

Proposition 24. *Let be a -hereditary ring and . Then, is a Kaplansky class.*

*Proof. *Let and . Consider the inclusion and we get by Lemma 5.3.12 in [13], a cardinal and a pure submodule such that and . We get the pure exact sequence
By Proposition 19, is -projective. It follows that
for any -injective -module This implies that is surjective by hypothesis. Thus, Hence, we proved the proposition.

Theorem 25. *Let be a -hereditary ring and . If the class of all -projective -modules is closed under direct limits, then, every -module has a -projective cover and an -injective envelope.*

*Proof. *By Proposition 24, is a Kaplansky class. Since all projective modules are -projective, contains the projective modules. Clearly, is closed under extensions. By hypothesis, is closed under direct limits. Then by Theorem 2.9 in [22], is a perfect cotorsion theory. Hence, by Definition 4, every module has a -cover and a -envelope.

Note that if then the class of all -projective -modules is closed under direct limits by Lemma 3.3.4 in [6]. Then, by Theorem 25 and Remark 22, we have the following.

Theorem 26. *Let be a -hereditary ring and . Then, every -module has a -projective cover and an -injective envelope.*

Now we introduce a self--injective ring.

*Definition 27. *A ring is said to be self -injective if over itself is an -injective module.

*Example 28. *(1)If the class is then is a self injective ring(2)If is the class of all finitely presented -modules, then is a self -injective ring(3)If is the class of all flat -modules, then is a cotorsion ring

We now give some characterizations of -projective module:

Proposition 29. *Let be a self -injective ring, and let be an -module. Then, the following conditions are equivalent:
*(1)* is -projective*(2)* is projective with respect to every exact sequence , with an -injective -module*(3)*For every exact sequence , where is -injective, is an -injective preenvelope of *(4)* is cokernel of an -injective preenvelope with is a projective -module*

*Proof. *. Let be an exact sequence, where is -injective. Then, by hypothesis, is surjective.

. Let be an -injective -module. Then, there is a sequence with an injective envelope of . By (2), is surjective. Thus, , as desired.

. Clearly, for all -injective . Hence, we have an exact sequence where is projective.

. Consider an exact sequence , where projective. Since is self--injective, every projective module is -injective. Hence, is -injective. Then, by hypothesis, is an -injective preenvelope.

. By hypothesis, there is an exact sequence , where is an -injective preenvelope with projective. It gives rise to the exactness of for each -injective -module . Since is self -injective, is surjective. Hence, , as desired.

Proposition 30. *Let be a self -injective and -hereditary ring and . Then the following are equivalent for an -module :
*(1)* is coreduced -projective*(2)* is a cokernel of an -injective envelope with is a projective -module*

*Proof. *. Consider an exact sequence with a projective module. Since is self -injective, is -injective. By Theorem 26, the natural map is an -injective preenvelope of . By hypothesis, has an -injective envelope . Then, there exist and such that and . Hence, . It follows that is an isomorphism, . Note that , and so is exact. But is coreduced and , and hence, , that is, So is an isomorphism, and hence, is an -injective envelope of .

. By Proposition 29, is -projective and is coreduced by Lemma 3.7 in [17].

We are now to prove the main result of this section.

Theorem 31. *Let be a self -injective and -hereditary ring and . Then, an -module is -projective if and only if is a direct sum of a projective -module and a coreduced -projective -module.*

*Proof. *The â€śifâ€ť part is clear.

â€śOnly ifâ€ť part. Let be a -projective -module. By Proposition 29, we have an exact sequence with a projective module, where is an -injective preenvelope of . By Theorem 26, has an -injective envelope with an -injective -module. Then, we have the following diagram (Figure 1):

Note that is an isomorphism, and so . Since , and are projective. Therefore, is a coreduced -projective module by Proposition 30. By the Five Lemma, is an isomorphism. Hence, we have , where . In addition, we get the commutative diagram (Figure 2).

Hence, .

#### 5. Some Relation between -Projective and -Injective Modules

In this section, we deals with -injective envelope of a module and -projective module for any class in .

Theorem 32. *Let be an -injective envelope. Then, is -projective, and hence, is -projective whenever is -projective.*

*Proof. *It follows from Lemma 2.1.2 in [25].

Theorem 33. *Let be a minimal generator of all -projective extensions of . Then, is an -injective envelope of .*

*Proof. *It follows from Theorem 2.2.1 in [25].

Let be a submodule of a module . Then, is called a -projective extension of a submodule if is -projective.

Recall that among all -projective extensions of , we call one of them a generator for (or a generator for all -projective extensions of ) if for any -projective extension of , then there is a diagram (Figure 3).

Furthermore, a generator is called minimal if for all the vertical maps are isomorphisms whenever are replaced by , , respectively.

Theorem 34. *Let be a -hereditary ring and Then, for an -module , there must be a minimal generator whenever has a generator.*

*Proof. *It follows from Theorem 2.2.2 in [25].

Theorem 35. *Suppose that an -module has an -injective envelope. Let be a submodule of an -injective -module . Then, the following are equivalent:
*(1)* is a special -injective envelope** is -projective, and there are no direct summands of with and *(2)* is -projective, and for any epimorphism such that is split, , where is the canonical map*(3)* is -projective, and any endomorphism of such that is a monomorphism*(4)* is -projective, and there is no nonzero submodule of such that and is -projective*

*Proof. *. It follows from Corollary ~1.2.3 in [25] and Theorem 32.

. Since is split, there is a monomorphism such that . Note that , and so by (2). Thus, , and hence, .

. If with . Let be a canonical projection. Then, there is an epimorphism such that . Thus, by hypothsis, and hence, , as required.

. By Wakamatsuâ€™s Lemma ([13], Proposition ~7.2.4), is -projective. Since and is monomorphism, is monomorphism.

. Since is -projective, is a special -injective preenvelope. Let be an -injective envelope of . Then, there exist and such that and . Hence and . Thus, is an isomorphism, and so is epic. In addition, by (4), is monic, and hence, is monic. Therefore, is an isomorphism, and hence, is an -injective envelope of .

. It is obvious that is -projective. Suppose there is a nonzero submodule such that and is -projective. Let be a canonical map. Since is -projective and is -injective, there is a in the following diagram (Figure 4)

Hence, . Note that is an envelope, and so is an isomorphism, whence is an isomorphism. But this is impossible since .

. Let be an -injective envelope of . Since is -projective, is a special -injective preenvelope. Thus, we have the following diagram shown in Figure 5.

That is, . So . Note that is an -injective envelope, and hence, is an isomorphism. Without loss of generality, we may assume . Write . It is clear that is epic and . We show that . Clearly, . Let . Then, . It follows that for some , and hence, . Thus, , and so , as desired. Consequently, is -projective by Wakamatsuâ€™s Lemma. Thus, by hypothesis, and hence, is an isomorphism. So is an -injective envelope.

Theorem 36. *Let be a -hereditary ring and If is a submodule of an -injective -module , then, the following are equivalent:
*(1)* is a special -injective envelope of ** is a -projective essential extension of *

*Proof. *. It follows by Proposition 35.

. By hypothesis, we have an exact sequence: with an -injective module and an