Covers and Envelopes by Submodules or Quotient-Modules

Zeng, Yuedi

doi:https://doi.org/10.1155/2023/9740245

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Research Article | Open Access

Volume 2023 | Article ID 9740245 | https://doi.org/10.1155/2023/9740245

Covers and Envelopes by Submodules or Quotient-Modules

Yuedi Zeng¹

Academic Editor: Faranak Farshadifar

Received26 Oct 2022

Revised10 Mar 2023

Accepted16 Mar 2023

Published15 Apr 2023

Abstract

Let be a ring, a class of left -modules, the class of submodules of , and the class of quotient-modules of . It is shown that is precovering (preenveloping) if and only if every injective (projective) left -module has an -precover (-preenvelope). Both epic and monic -(pre) covers (-(pre) envelopes) are studied. Moreover, some applications are given. In particular, it is proven that the injective envelope of any projective left -module is projective if and only if the class of quotient-modules of projective and injective left -modules is monic preenveloping.

1. Introduction

Throughout this paper, is an associative ring with identity, and modules are unitary. For a left -module , stands for the injective envelope. The character module is defined by . The class of projective (injective) left -modules is denoted by .

Let be a class of left -modules and a left -module. Following [1], we say that a homomorphism is a -preenvelope of if and the abelian group homomorphism Hom : Hom Hom is surjective for each . A -preenvelope is called a -envelope if every endomorphism such that is an isomorphism. Dually, we have the definitions of -precovers and -covers. -envelopes (-covers) may not exist in general, but if they exist, they are unique up to isomorphisms. Hence, we will always assume that the classes of left -modules are closed under isomorphisms in this paper.

Note that the class of submodules of injective left -modules or the class of quotient-modules of projective left -modules is the class of left -modules. It is clear that this class is precovering (preenveloping). Hence, this paper is motivated by the following questions:

Let be a class of left -modules.

Question 1. When is the class of submodules of (pre) covering?

Question 2. When is the class of quotient-modules of (pre) enveloping?
In Section 2, it is shown that is precovering if and only if every injective left -module has an -precover. It is also proven that is epic precovering if and only if every injective left -module has an epic -precover. There are many applications. It is shown that:(1)The class of submodules of projective left -modules is epic precovering.(2)The class of submodules of flat left -modules is epic precovering.(3)Let be a right coherent ring. The class of submodules of pure-injective flat left -modules is epic precovering.(4)Let be a commutative noetherian ring. The class of submodules of flat-cotorsion -modules is epic precovering.It is also proven that if any injective left -module has a projective cover (e.g., the ring is a perfect ring), then the following are equivalent:(1)The class of submodules of projective and injective left -modules is epic precovering.(2)The projective cover of any injective left -module is injective.Moreover, suppose that the class is closed under pure submodules, direct products, and direct limits, the class of submodules of is covering if and only if every injective left -module has an -cover (see Theorem 2). There are many examples. It is proven that:(1)The class of submodules of -flat left -modules is epic covering.(2)If is right -coherent, then the class of submodules of -flat left -modules is epic covering.In Section 3, it is shown that is preenveloping if and only if every projective left -module has an -preenvelope. It is also shown that is monic preenveloping if and only if every projective left -module has a monic -preenvelope. There are many corollaries. It is proven that:(1)The class of quotient-modules of injective left -modules is monic preenveloping.(2)The class of quotient-modules of pure-injective left -modules is monic preenveloping.(3)The class of quotient-modules of FP-injective left -modules is monic preenveloping.(4)The class of quotient-modules of -injective left -modules is monic preenveloping.(5)The class of quotient-modules of -injective left -modules is monic preenveloping.(6)Let be a right coherent ring. Then the class of quotient-modules of pure-injective flat left -modules is monic preenveloping.It is well known that each module has injective envelope. It is also proven that the injective envelope of any projective left -module is projective if and only if the class of quotient-modules of projective and injective left -modules is monic preenveloping.

2. Precovers by Submodules

In this section, we study Question 1.

Lemma 1. Let be a class of left -modules, the class of submodules of , and a left -module. If E has an -precover , then has an -precover with .

Proof. Let be the injective envelope (we may regard as the inclusion). Set . Then, there is a morphism such that the following diagram commutes:
where is the inclusion. Thus, with and is the pullback for and by [[2], Chap IV, §5]. For any left -module with and any homomorphism , there is a morphism such that the following diagram commutes:
where is the inclusion map. Moreover, there exists a morphism such that since is an -precover. Thus, . By the factorization over (see [[2], Chap IV, §5]), there is a left -homomorphism (, ) such that and . Hence, there exists the following commutative diagram with exact rows:

Therefore, is an -precover of .

Theorem 1. Let be a class of left -modules and the class of submodules of . The following are equivalent:(1)The class is precovering(2)Every injective left -module has an -precover(3)Every injective left -module has an -precover with (4)Every injective left -module has an -precover

Proof. (1) (2) and (3) (2) are trivial (2) (3) Let be any injective left -module and an -precover of . Note that . There is a left -module such that is a submodule of . Since is injective, there is a morphism such that the following diagram commutes:where is the canonical inclusion. Obviously, . Then, is an -precover of . (3) (4) Since and , is an -precover of (4) (1) follows by Lemma 1

Corollary 1. Suppose that the class is closed under submodules. Then, is precovering if and only if every injective left -module has an -precover.

Recall that a torsion theory (see [2], 2) for left -modules consists of two classes and , the torsion class and the torsion-free class, respectively, such that Hom , whenever and . Then, the class is closed under quotient-modules, extensions, and direct sums (see [2], Proposition 2.1), and the class is closed under submodules, extensions, and direct products (see [2], Proposition 2.2).

Example 1. Let be a torsion theory. Then, is precovering if and only if every injective left -module has an -precover.

A torsion theory is called hereditary (see [2], 3) if is closed under submodules.

Example 2. Let be a hereditary torsion theory. Then, is precovering if and only if every injective left -module has a -precover.

Now, we consider epic precover in Theorem 1. If every left -module has an epic -(pre) cover, we write that is epic (pre) covering.

Lemma 2. Let be a class of left -modules, the class of submodules of , and a left -module. If E has an epic -precover , then has an epic -precover such that .

Proof. In view of the proof of Lemma 1 and [[2], Chap IV, Proposition 5.1], there is a commutative diagram with exact rows

where is the injective envelope of and the right square is a pullback diagram. Thus, is an epic -precover of by Lemma 1.

Proposition 1. Let be a class of left -modules and the class of submodules of . The following are equivalent:(1)Every left -module has an epic -precover(2)Every injective left -module has an epic -precover(3)Every injective left -module has an epic -precover with (4)Every injective left -module has an epic -precover.

Proof. (1) (2), (3) (2), and (3) (4) are trivial (2) (3). Let be any injective left -module and be an epic -precover of . Note that . There is a left -module such that is a submodule of . Since is injective, there is a morphism such that the following diagram commutes:where is the canonical inclusion. Since is epic, is epic too. It follows that is an epic -precover of since . (4) (1) follows by Lemma 2.

Example 3. (1)The class of submodules of projective left -modules is epic precovering(2)The class of submodules of flat left -modules is epic precovering.

Proof. (1)Obviously, any module has an epic projective precover. So, (1) follows from Lemma 2(2)By [[3], Theorem 3], any module has an epic flat cover. So (2) follows from Lemma 2.Set = {flat -modules} {cotorsion -modules}. The flat-cotorsion class has been studied by many authors ([4–7] etc.).

Example 4. Let be a commutative noetherian ring. Then, the class of submodules of flat-cotorsion -modules is epic covering.

Proof. Let be an injective -module. Then has an epic flat cover by [3]. It follows that is flat-cotorsion by [[4], Theorem 5.3.28]. For any flat-cotorsion -module and any homomorphism , there is a morphism such that . Thus, is an epic flat-cotorsion cover of . The result follows from Lemma 2.
Note that any pure-injective left -module is cotorsion. Set = {flat left -modules} {pure-injective left -modules}.
Recall that a ring is said to be right coherent (see [8]) in case each finitely generated right ideal of is finitely presented. We have the following.

Example 5. Let be a right coherent ring. Then, the class of submodules of pure-injective flat left -modules is epic precovering.

Proof. Let be an injective -module. By [[4], Theorem 5.3.11], we have that every injective left -module has a flat cover with flat and pure-injective. Clearly, is an epimorphism. Thus, is an epic pure-injective flat cover of . The result follows from Lemma 2.

Corollary 2. Let be a class of left -modules such that , and be the class of submodules of . If any injective left -module has a projective cover (e.g., the ring is a left perfect ring) and is closed under direct summands, then the following are equivalent:(1)Every left -module has an epic -precover(2)Every injective left -module has an epic -precover(3)Every injective left -module has an epic -precover with (4)Every injective left -module has an epic -cover with (5)Every injective left -module has an epic -precover(6)Every injective left -module has an epic -cover(7)The projective cover of any injective left -module is in .

Proof. (1) (2) (3) (5) follow from Proposition 1 (4) (3) and (6) (5) are trivial (7) (6) Let be an injective left -module and be the projective cover of . Clearly, is epic. By (7), .Because , is an epic -cover of . (5) (6), (7) Let be an injective left -module, an epic -precover of and the projective cover of . Note that and are both projective. There exist morphism and such that and . Hence, . Since is a cover, is an isomorphism. Thus, is a direct summand of . It follows that is in . Thus, is an epic -cover. (6) (4) Let be an injective left -module and an epic -cover of . Note that (6) (3). Then, there is an epic -precover with . And so there is a morphism such that . Thus, is an epic -cover of .Let = {injective left -modules} {projective left -modules}.

Example 6. If any injective left -module has a projective cover, then the following are equivalent.(1)The class of submodules of projective and injective left -modules is epic precovering.(2)The projective cover of any injective left -module is injective.

Proof. (1) (2). Let be an injective left -module and be the projective cover of . By Corollary 2, we get that = {injective left -modules} {projective left -modules}. Thus, is injective. (2) (1) is trivial by Corollary 2.A left -module is called FP-injective (or absolutely pure) [9, 10] if for all finitely presented left -modules . Let = {FP-injective left -modules} {projective left -modules}.

Example 7. If any injective left -module has a projective cover, then the following are equivalent:(1)The class of submodules of projective and FP-injective left -modules is epic precovering.(2)The projective cover of any injective left -module is FP-injective.

Next, we consider the monic precover.

Lemma 3. Let be a class of left -modules, the class of submodules of , and a left -module. If E has a monic -cover , then has a monic -cover with .

Proof. According to the proof of Lemma 1, we get that with and is the pullback:

It follows that is monic by [[2], Chap IV, Proposition 5.1(i)]. Thus, is a monic -precover of by Lemma 1.
The following example shows that the necessary and sufficient conditions for epic -precover (in Theorem 1) do not apply to monic -(pre) cover.

Example 8. Let be a semisimple ring. If , where and are two nonisomorphic simple left modules. Now, let and be the class of submodules of . Since is semisimple, every left -module has a monic -precover by [[11], Proposition 13.9]. Note that is injective. But has an -precover , where is the canonical projection. And, monic -cover of does not exist.

Finally, we consider when is the class of submodules of covering.

Lemma 4. Suppose that the class is closed under pure submodules, direct products, and direct limits. Then, the class of submodules of is closed under direct limits.

Proof. The proof is similar to the proof of [[4], Lemma 5.3.12].
Let be a well ordered inductive system with each a submodule of a left -module . We need to show that is also a submodule of a left -module in .
By [[4], Lemma 5.3.12], there is a cardinal number (dependent on Card and Card ) such that if is any morphism with , then there is a pure submodule with and Card . Note that is closed under pure submodules, then .
Let be a set of { and Card }. For each , we consider all morphisms with . Let , over all such , and be the morphism . Then, since is closed under direct products. Note that is a submodule of a left -module . There is a monic morphism . Hence, there is a left -module such that is the inclusion. So, is an injection.
Let (over morphisms described above). If is a morphism, the decomposition (the last map being the projection map) is one of the morphisms , that is, and is the morphism . So, let be the projection map corresponding to . Then, we see that
is commutative and the morphisms are functorial in the obvious sense. So, we can define an direct limit . Note that is an injection. So, is also an injection. Thus, we are done since .
From [[4], Corollary 5.2.7] and Theorem 1, we get the following theorem immediately.

Theorem 2. Suppose that the class is closed under pure submodules, direct products, and direct limits. The class of submodules of is covering if and only if every injective left -module has an -cover.

As applications, we have the following examples.

Recall that a left -module is said to be fp-flat [12] if for every monomorphism with and finitely presented right -modules, is a monomorphism. A left -module is said to be fp-injective [12] if for every monomorphism with and finitely presented left -modules, Hom is an epimorphism.

Lemma 5 (see [13]). Theorem 3.8 and Proposition 3.11. (1)The class of -flat left -modules is closed under direct products, direct sums, direct summands, and direct limits.(2)The class of -injective left -modules is closed under direct products, direct sums, direct summands, and direct limits.

Lemma 6 [(see [13]), Theorem 3.3]. A left -module is fp-injective (fp-flat) if and only if is fp-flat (fp-injective).

Corollary 3. (1)The class of -flat left -modules is closed under pure submodules and pure quotient-modules.(2)The class of -injective left -modules is closed under pure submodules and pure quotient-modules.

Proof. Let be a pure exact sequence. This induces a split exact sequence: . By Lemma 6, if is -flat (-injective), then is -injective (-flat). This means that and are -injective (-flat) by Lemma 5. Thus, and are -flat (-injective) by Lemma 6. Hence, the result follows.

Proposition 2. (1)The class of -flat left -modules is covering(2)The class of -injective right -modules is covering

Proof. Clearly, the class of -flat left -modules or the class of -injective right -modules is closed under direct sums and pure quotient modules by Lemma 5 and Corollary 3. Thus, the result follows from [[14], Theorem 2.5].

Example 9. The class of submodules of -flat left -modules is epic covering.

Proof. It follows from Lemma 5, Proposition 2, and Theorem 2.

Lemma 7 [(see [12]), Theorem 2.4]. A ring is right coherent if and only if every fp-flat left -module is flat.

Example 10. [(see [4]), Theorem 5.3.14]. If is right coherent, then the class of submodules of flat left -modules is epic covering.

Let and be fixed positive integers. A right -module is said to be -presented [15] if there exists an exact sequence of right -modules, where is -generated. A ring is called right -coherent [15] in case each -generated submodule of the right -module is finitely presented. A right -module is said to be -injective [16] if for any -presented right -module ; a left -module is said to be -flat [15] if for any -presented right -module . From the definitions, it is easy to see that:

FP-injective = -injective for all positive integers and ,

Flat = -flat for all positive integers = -flat for all positive integers and ,

Coherent = -coherent for all positive integers = -coherent for all positive integers and .

A ring is said to be right J-coherent [17] if is a coherent right -module, where is the Jacobson radical of . A left -module is said to be J-flat [17] if for every finitely generated right ideal in . A right -module is called J-injective [17] if Ext for every finitely generated right ideal in .

A ring is said to be right N-coherent [18] if is a coherent left -module, where is the intersection of all prime ideals of . A left -module is said to be N-flat [18] if for every finitely generated right ideal in . A right -module is called J-injective [17] if for every finitely generated right ideal in .

Remark 1. By definitions, the class of -flat (-flat, -flat) left -modules is closed under direct limits, direct summands, direct sums, pure submodules, and pure quotient-modules. Then every left -module has an epic -flat (-flat, -flat) cover by [[14], Theorem 2.5]. Hence, the class of submodules of -flat (-flat, -flat) left -module is precovering by Theorem 1.
If is right -coherent (-coherent, -coherent), then the class of -flat (-flat, -flat) left -modules is closed under direct product (see [[15], Theorem 5.6], [[17], Theorem 2.13], [[18], Theorem 2.13]). Hence, the class of submodules of -flat (-flat, -flat) left -modules is covering by Theorem 2.

3. Preenvelopes by Quotient-Modules

In this section, we study Question 2.

Lemma 8. Let be a class of left -modules, the class of quotient-modules of , and a left -module. If is a projective resolution of and has an -preenvelope , then has a -preenvelope.

Proof. Let , , and be the natural epimorphism. Then there is a homomorphism such that the following diagram commutes:

Note that is projective. For any epimorphism with and any homomorphism , there is a morphism such that the following diagram commutes:

Moreover, there exists a morphism such that since is an -preenvelope. Thus, . This implies that . Hence, there is a induced morphism such that . This means that . Since is epic, . And so we have the following commutative diagram:

It follows that is a -preenvelope of .

Theorem 3. Let be a class of left -modules and the class of quotient-modules of . The following are equivalent:(1)The class is preenveloping.(2)Every projective left -module has a -preenvelope.(3)Every projective left -module has a -preenvelope with .(4)Every projective left -module has an -preenvelope.

Proof. (1) (2) and (3) (2) are trivial. (2) (3) Let be any projective left -module and be a -preenvelope of . Note that , there is an epimorphism with . Since is projective, there is a morphism such that . Clearly, . It follows that is a -preenvelope of with . (3) (4) Since and , is an -preenvelope of . (4) (1) follows from Lemma 8.

Corollary 4. Suppose that the class is closed under quotient-modules. Then, is preenveloping if and only if every projective left -module has an -preenvelope.

Example 11. Let be a torsion theory. Then, is preenveloping if and only if every projective left -module has a -preenvelope.

If every left -module has a monic -(pre) envelope, we write that is monic (pre) enveloping.

Note that

In Lemma 8 is the pushout for and . Dual to Lemma 2, we get the following.

Lemma 9. Let be a class of left -modules, the class of quotient-modules of , and a left -module. If is a projective resolution of and has a monic -preenvelope , then has a monic -preenvelope.

Proposition 3. Let be a class of left -modules and the class of quotient-modules of . The following are equivalent:(1)Every left -module has a monic -preenvelope(2)Every projective left -module has a monic -preenvelope(3)Every projective left -module has a monic -preenvelope with (4)Every projective left -module has a monic -preenvelope.

Proof. (1) (2), (3) (2), and (3) (4) are trivial (2) (3) follows from Theorem 3 (4) (1) follows from Lemma 9

Example 12. (1)The class of quotient-modules of injective left -modules is monic preenveloping(2)The class of quotient-modules of pure-injective left -modules is monic preenveloping(3)The class of quotient-modules of FP-injective left -modules is monic preenveloping(4)The class of quotient-modules of -injective left -modules is monic preenveloping(5)The class of quotient-modules of -injective left -modules is monic preenveloping(6)The class of quotient-modules of -injective left -modules is monic preenveloping(7)The class of quotient-modules of -injective left -modules is monic preenveloping

Proof. (1)Obviously, any module has a monic injective envelope. So, (1) follows from Lemma 9.(2)By [[4], Example 6.6.5], we get that any left -module module has a monic pure-injective envelope. So, (2) follows from Lemma 9.(3)By [[4], Theorem 6.2.4], we get that any left -module module has a monic FP-injective preenvelope. So, (3) follows from Lemma 9.(4)By Lemma 5, Corollary 3 and [[4], Lemma 5.3.12, and Theorem 6.1.2], we get that any left -module has a monic -injective preenvelope. So, (4) follows from Lemma 9.(5)By [[19], Theorem 3.1], we get that any module has a monic -injective preenvelope. So, (5) follows from Lemma 9.(6)By [[17], Lemma 2.4] and [[4], Lemma 5.3.12, and Theorem 6.1.2], we get that any module has a monic -injective preenvelope. So, (6) follows from Lemma 9.(7)By [[18], Remark 3.11], we get that every left -module has a monic -injective preenvelope. So, (7) follows from Lemma 9.Set = {pure-injective left -modules} {flat left -modules}.

Example 13. Let be a right coherent ring. Then, the class of quotient-modules of pure-injective flat left -modules is monic preenveloping.

Proof. Let be a projective left -module. There is a monic pure-injective envelope with pure-injective by [[4], Example 6.5.5(2)]. It follows that is flat by [[4], Proposition 6.7.1]. For any pure-injective flat left -module and any homomorphism , there is a morphism such that . Thus, is a monic pure-injective flat envelope of . The result follows from Lemma 9.

Corollary 5. Let be a class of left -modules and be the class of quotient-modules of . If is closed under direct summands and , then the following are equivalent:(1)Every left -module has a monic -preenvelope(2)Every projective left -module has a monic -preenvelope(3)Every projective left -module has a monic -preenvelope with (4)Every projective left -module has a monic -envelope with (5)Every projective left -module has a monic -preenvelope(6)Every projective left -module has a monic -envelope(7)The injective envelope of any projective left -module is in .

Proof. (1) (2) (3) (5) follow from Proposition 3. (4) (3) and (6) (5) are trivial. (7) (6) Let be a projective left -module and be the injective envelope of . Clearly, is monic. By (7), . Because by hypothesis, is a monic -envelope of . (5) (6), (7) Let be a projective left -module, a monic -preenvelope of and the injective envelope of . Note that and are both injective. Then there exist morphism and such that and . Hence, . Since is an envelope, is an isomorphism. Thus, is a direct summand of . It follows that is in . Thus, is a monic -envelope. (6) (4) Let be a projective left -module and be a monic -envelope of . Since (6) (3), there is a monic -preenvelope with . This implies that there is a morphism such that . Thus, is a monic -enveloper of . Let = {injective left -modules} {projective left -modules}.

Corollary 6. The following are equivalent:(1)The class of quotient-modules of projective and injective left -modules is monic preenveloping.(2)The injective envelope of any projective left -module is projective.

Proof. (1) (2) Let be a projective left -module and be the injective envelope of . By Corollary 5, we get that (={injective left -modules} {projective left -modules}). Thus, is projective. (2) (1) is trivial by Corollary 5.Let = {injective left -modules} {flat left -modules}.

Corollary 7. The following are equivalent:(1)The class of quotient-modules of flat and injective left -modules is monic preenveloping.(2)The injective envelope of any projective left -module is flat.

Dual to Lemma 3, we get the following.

Lemma 10. Let be a class of left -modules, the class of quotient-modules of , and a left -module. If is a projective resolution of and has an epic -envelope , then has an epic -envelope.

The following example shows that the necessary and sufficient conditions for monic -precover (Proposition 3) do not apply to epic -preenvelope.

Example 14. Let be a semisimple ring. If R = , where and are two nonisomorphic simple left modules. Now, let and be the class of quotient-modules of . Since is semisimple, every left -module has an epic -preenvelope by [[11], Proposition 13.9]. Note that is projective. But has an -preenvelope , where is the canonical injection. And, epic -preenvelope of does not exist.

Remark 2. It would be interesting to study pure-submodules. Let PE be the pure-injective envelope of . According to the proof of Lemma 1, we may get the following Proposition.
Let be a class of left -modules, the class of pure-submodules of , and a left -module. If PE has an -precover PE , then has an -precover with pure in .
Therefore, we can get the corresponding results on precovers by pure-submodules. Preenvelopes by pure-quotient-modules may also be studied dually.

Data Availability

The data supporting the results of this study and the PDF file are the same.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The work was supported by Natural Science Foundation of Fujian Province, China (2020J01908 and 2020J05206).

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Copyright © 2023 Yuedi Zeng. 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.

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