<span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2723999pt" id="M1" height="11.6718pt" version="1.1" viewBox="-0.0657574 -11.3994 8.2614 11.6718" width="8.2614pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g113-84" d="M449 634C442 637 425 643 405 650C376 660 341 666 307 666C181 666 98 590 98 485C98 400 170 343 215 310L246 288C307 243 343 204 343 147C343 67 291 18 219 18C104 18 61 124 51 202L23 199C28 124 27 71 27 47C47 22 122 -16 204 -16C324 -16 428 60 428 174C428 256 379 309 307 360L276 382C223 419 179 455 179 516C179 576 221 632 293 632C379 632 410 564 418 487L448 490C446 536 446 592 449 634Z"/></g></svg>-</span>FP-Projective Modules and Dimensions

Assaad, Refat Abdelmawla Khaled; Zhang, Xlaolei; Kim, Hwankoo

doi:https://doi.org/10.1155/2023/7151101

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Research Article | Open Access

Volume 2023 | Article ID 7151101 | https://doi.org/10.1155/2023/7151101

-FP-Projective Modules and Dimensions

Refat Abdelmawla Khaled Assaad,¹Xlaolei Zhang,²and Hwankoo Kim³

Academic Editor: Marco Fontana

Received26 Jan 2023

Revised12 Feb 2023

Accepted04 Apr 2023

Published25 Apr 2023

Abstract

Let be a ring and let be a multiplicative subset of . An -module is said to be a --absolutely pure module if is --torsion for any finitely presented -module . This paper introduces and studies the notion of -FP-projective modules, which extends the classical notion of FP-projective modules. An -module is called an -FP-projective module if for any --absolutely pure -module . We also introduce the -FP-projective dimension of a module and the global -FP-projective dimension of a ring. Then, the relationship between the -FP-projective dimension and other homological dimensions is discussed.

1. Introduction

Throughout the paper, all rings considered are commutative with identity, all modules are unitary, and is always a multiplicative subset of , that is, and for any . A multiplicative subset of is said to be finite if the cardinal of is finite. Let be a ring and be an -module. As usual, we use , , and to denote the classical injective dimension, projective dimension, and flat dimension of , respectively, and and to denote the global and weak homological dimensions of , respectively. We also use “f.g.” (resp., “f.p.”) as shorthand for “finitely generated” (resp., “finitely presented”).

From reference [1], we recall that an -module is said to be --torsion if for some . An -module is said to be -finite if is --torsion for some f.g. submodule of . Also, following Zhang [1, 2], a sequence is said to be --exact (at ) provided that there is an element such that and . A long -sequence is called --exact if for any , there is an element such that and . A --exact sequence is called a short --exact sequence. A homomorphism is a --monomorphism (resp., --epimorphism and --isomorphism) provided that (resp., and ) is --exact. It is easy to verify that a homomorphism is a --monomorphism (resp., --epimorphism and --isomorphism) if is (resp., is, both and are) --torsion.

Maddox [3] called a module absolutely pure if it is pure in every module containing it as a submodule. In reference [4], Megibben showed that an -module is absolutely pure if and only if for every f.p. -module . Thus, an absolutely pure module is called an FP-injective module in [5]. Recently, the concept of --absolutely pure modules (abbreviates uniformly -absolutely pure) is introduced in reference [6] as a generalization of that of absolutely pure modules. As in reference [6], a --exact sequence of -modules is called --pure provided that for every -module , the induced sequence is also --exact, and a submodule of is called a --pure submodule if the exact sequence is --pure exact. An -module is said to be --absolutely pure provided that any short --exact sequence beginning with is --pure. By reference [6], Theorem 3.2, an -module is --absolutely pure if and only if there exists an element , satisfying that is --torsion with respect to for any f.p. -module .

Recently, Zhang in reference [1] defined the --von Neumann regular ring as follows: A ring is called a --von Neumann regular ring if there exists an element such that for any , there exists with . Thus, by reference [1], Theorem 3.13, is a --von Neumann regular ring if and only if every -module is --flat, and in reference [6] Theorem 3.5, it is proved that a ring is --von Neumann regular if and only if every -module is --absolutely pure.

In reference [7], the authors introduced and characterized the concept of the FP-projective dimension of modules and rings. The -projective dimension of an -module is the smallest integer such that for any absolutely pure -module . The -projective dimension of is defined as the supremum of the -projective dimensions of the f.g. -modules. These dimensions measure how far an f.g. module is from being f.p. and how far away a ring is from being Noetherian, respectively. For example, they proved that a ring is Noetherian if and only if every -module is FP-projective. Recall that is called a hereditary ring (resp., an FP-hereditary ring) if every ideal of is projective (resp., FP-projective) (see, ([8], Definition 3.1)). It is trivial that the projective module is FP-projective, and so, the hereditary ring is FP-hereditary. A natural question is whether a new class of modules (resp., rings) exists between the classes of these two modules (resp., rings). From this point of view, in reference [9], -FP-projective modules and dimensions were introduced and studied using the torsion theory derived from the star operation . The motivation of this paper is to unify these concepts in the module case and the ring case using a multiplicative subset of the ring.

Section 2 introduces the concept of -FP-projective modules and gives some characterizations of -FP-projective modules. Using these results, we prove that a ring is coherent if and only if every ideal of is -FP-projective; if and only if every f.g. submodule of a projective module is -FP-projective. Also, we prove that is an -FP-hereditary ring if and only if every submodule of a projective -module is -FP-projective; if and only if every submodule of an -FP-projective -module is -FP-projective.

Section 3 deals with the -FP-projective dimension of a module , denoted by -, and the global -FP-projective dimension of a ring , denoted by . Among other results, we characterize when - and when , as is usually carried out in the study of the classical homology dimensions. In particular, it is shown that , if and only if every submodule of projective (resp., --projective) -module is --projective; if and only if for any --absolutely pure -module ; if and only if is an -FP-hereditary ring. Finally, a nontrivial example that FP-hereditary rings are not -FP-hereditary, in general, is given.

2. -FP-Projective Modules

In this section, we introduce a class called the -FP-projective module, study their properties, and characterize them. We begin this section with the following definition:

Definition 1. An -module is called -FP-projective if for any --absolutely pure -module .
Since every absolutely pure module is --absolutely pure by reference [6], Proposition 3.3, we have the following implications:

Remark 1. (1)If consists of units, then --absolutely pure modules and absolutely pure modules coincide. Thus, the -FP-projective -modules are just the FP-projective -modules.(2)If , then every -module is --absolutely pure. So, -FP-projective modules are exactly projective modules.(3)Using reference [4], Theorem 5, it is easy to see that the three classes of modules previous coincide over a von Neumann regular ring.In the following example, we show that there exists an FP-projective -module but not -FP-projective.

Example 1. Let , the ring of integers, a prime in , and . Since is Noetherian, all -modules are FP-projective by ([7], Proposition 2.6). Since is --torsion, it is also --absolutely pure (see ([6], Example 3.8)). However, since (see ([10], page 267)), is not -FP-projective.
Now, we characterize rings over which all -FP-projective modules are projective.

Proposition 1. Let be a ring. Then, is --von Neumann regular if and only if all -FP-projective modules are projective.

Proof. Suppose that is --von Neumann regular. Then, all -modules are --absolutely pure by reference [6], Theorem 3.5. So, all -FP-projective modules are projective. The sufficiency follows similarly. □
We know that every f.g. projective -module is f.p.; we generalize this result to -FP-projective modules in the following proposition:

Proposition 2. Let be a ring. Then, every f.g. -FP-projective -module is f.p.

Proof. Let be an f.g. -FP-projective -module. Then, for any --absolutely pure module . Hence, it follows by reference [11], Theorem 2.1.10 that is f.p. □
Now, we give some characterizations of -FP-projective modules.

Proposition 3. Let be a ring and be an -module. Then, the following are equivalent:(1)is-FP-projective(2) is projective with respect to every exact sequence , where is--absolutely pure(3)For any exact sequence of -modules of the form and any --absolutely pure module , the sequence is exact(4)Every exact sequence of the form , where is--absolutely pure, splits(5)is-FP-projective for any projective -module (6) is -FP-projective for any f.g. projective -module

Proof. Let be an exact sequence with --absolutely pure. Then, we have the exact sequence . Since is -FP-projective and is --absolutely pure, . Thus, is exact.
Let M be a --absolutely pure module. Consider the following exact sequence with an injective module. So, we have the following exact sequence:Keeping in mind that is exact, we deduce that . Hence, is -FP-projective.
Let be an exact sequence. For any --absolutely pure module , it follows that is exact. Since is -FP-projective, , and so (3) holds.
Let be an exact sequence with projective. Hence, for any --absolutely pure , we have the following exact sequence:Since is exact, we deduce that . Hence, is -FP-projective.
It is clear.
Let be a --absolutely pure -module and be a projective -module. By ([12], Theorem 3.3.10), we have the following isomorphism:Thus, since is -FP-projective. Hence, , and so is -FP-projective.
Let be a --absolutely pure -module and be an f.g. projective -module. By reference [12], Theorem 3.3.12, we have the following isomorphism:Thus, is an -FP-projective -module.
and These follow by setting . □
Recall that an -module is said to be -torsion if for any , there exists such that .

Lemma 1. Let be a ring and be finite. Then, every -torsion -module is --absolutely pure.

Proof. Let be an -module and be an f.p. -module. Then, the natural homomorphisminduces a homomorphismBy reference [13], Proposition 1.10, is a monomorphism. Let be an -torsion -module. Then, since by reference [12], Example 1.6.13. Hence, is -torsion by reference [12], Example 1.6.13 again. Hence, is --torsion by ([1], Proposition 2.3). Consequently, is a --absolutely pure by reference [6], Theorem 3.2. □
The following proposition gives a condition that all -FP-projective modules are projective:

Proposition 4. Let be a ring, be finite, and be an -module. If is -FP-projective and for any -torsion-free -module , then is projective.

Proof. Let be an -module. The exact sequence,gives rise to the following exact sequence:The left term is zero by Lemma 1 and the right term is zero since is -torsion-free (see ([12], Example 1.6.13)). Thus, , and so is projective. □
Recall that a ring is said to be coherent if every f.g. ideal of is f.p.

Lemma 2. Let be a coherent ring, be finite and be an -module. Then, the following conditions are equivalent:(1)is--absolutely pure(2)There exists an element satisfying that for any f.p.-module and any integer ,is--torsion with respect to

Proof. Suppose that is a --absolutely pure -module and let be an f.p. -module. The case is trivial by ([6], Theorem 3.2). Thus, we may assume that . Consider an exact sequence as follows:where is an f.p., and free -module and are f.g.. Such sequence exists because is coherent. Hence, we have the following isomorphism:by reference [12], Example 1.6.13 since every --torsion is -torsion. Thus, , which implies that is an -torsion -module by reference [12], Example 1.6.13 and is --torsion by reference [1], Proposition 2.3.
This is obvious. □

Lemma 3. Let be a coherent ring, be finite, and be an exact sequence of -modules, where is --absolutely pure. Then, is --absolutely pure if and only if is --absolutely pure.

Proof. Let be an f.p. -module. We have the following exact sequence:By Lemma 2, ([1], Proposition 2.3), and ([12], Example 1.6.13), we have the following exact sequence:Hence, . So, is an -torsion -module if and only if is an -torsion -module by ([12], Example 1.6.13). By ([1], Proposition 2.3), is a --torsion -module if and only if is a --torsion -module. Thus, is --absolutely pure if and only if is --absolutely pure. □

Proposition 5. Let be a coherent ring, be finite, and be an -module. Then, the following conditions are equivalent:(1) is -FP-projective(2) for any --absolutely pure -module and any integer

Proof. Let be a --absolutely pure -module. The case is trivial. So, we may assume . We consider the following exact sequence:where are injective -modules. By Lemma 3, is --absolutely pure. Hence, .
This is trivial. □

Proposition 6. Let be a coherent ring, be finite, and be an exact sequence of -modules, where is -FP-projective. Then, is -FP-projective if and only if is -FP-projective.

Proof. Let be a --absolutely pure -module. Then, we have the following exact sequence:Since is -FP-projective, , and by Proposition 5 we have . Thus, . Hence, is -FP-projective if and only if is -FP-projective. □

Proposition 7. Let be a ring, be finite, and be an exact sequence of -modules. If and are -FP-projective, then is -FP-projective.

Proof. For any --absolutely pure -module , we have the following exact sequence . Since and are -FP-projective, we have , and so . Therefore, is -FP-projective. □

Proposition 8. Let be finite. Then, the class of all -projective modules is closed under arbitrary direct sums and under direct summands.

Proof. It follows by ([12], Theorem 3.3.9(2)). □

Proposition 9. Let be a ring. If every --absolutely pure -module has injective dimension , then is a coherent ring.

Proof. Let be an f.g. ideal of and be a --absolutely pure -module. Then, by hypothesis, . Consider the following exact sequence:Hence, since . Thus, is -FP-projective. Then, by Proposition 2, is f.p., which implies that is a coherent ring. □
We recall from reference [5] that the FP-injective dimension of , denoted by -, is defined to be the least nonnegative integer such that for any f.p. -module .

Proposition 10. Let be a ring. We consider the following conditions:(1) is a coherent ring(2)Every f.g. submodule of a projective -module is -FP-projective(3)Every f.g. ideal of is -FP-projectiveThen, , and if is composed of units, we have .

Proof. It is obvious.
Let be an f.g. ideal of . Then, is -FP-projective by (3). Hence, is f.p. by Proposition 2. Thus, is coherent.
Let be an f.g. submodule of a projective -module . Hence, by ([9], Theorem 3.7), we have is absolutely pure (FP-injective), and so -FP-projective since is composed of units. □
In the following definition, we define the -FP-hereditary ring, which is an extension of the FP-hereditary ring.

Definition 2. A ring is said to be -FP-hereditary if every ideal of is -FP-projective.
Note that every -FP-hereditary ring is FP-hereditary since every -FP-projective module is FP-projective. Therefore, we have the following implications:

Remark 2. (1)If is composed of units, then the class of -FP-hereditary rings and the class of FP-hereditary rings coincide.(2)If , then every -FP-projective module is projective. Hence, all -FP-hereditary rings are exactly hereditary rings.Later, we will provide an example of an FP-hereditary ring but not -FP-hereditary (Example 2).

Lemma 4. Let be a ring. If every submodule of an -FP-projective -module is absolutely pure, then every FP-projective module is -FP-projective.

Proof. Let be an FP-projective -module. Then, there exists an exact sequence , where is an -FP-projective -module. Hence, by hypothesis, is absolutely pure, and so by ([9], Proposition 3.3), the exact sequence splits. Hence, by Proposition 3, is -FP-projective since every absolutely pure is --absolutely pure. □

Corollary 1. If is an FP-hereditary ring and any submodule of an -FP-projective -module is absolutely pure, then is -FP-hereditary.

Proof. Let be an ideal of . Then, is FP-projective since is FP-hereditary. Hence, is -FP-projective by Lemma 4, which implies that is -FP-hereditary. □
In the following result, we characterize -FP-hereditary rings.

Proposition 11. The following conditions are equivalent for a ring :(1)is-FP-hereditary(2)Every submodule of a projective -module is -FP-projective(3)Every submodule of an -FP-projective -module is -FP-projective(4)Every --absolutely pure -module has injective dimension(5)For any --pure submodule of an injective module , the factor module is injective

Proof. These are obvious.
Let be a --absolutely pure -module and be an ideal of . The exact sequence gives the following exact sequence:Thus, , which implies that .
Let be a submodule of an -FP-projective -module . By (4), for any --absolutely pure -module , we have the following exact sequence:where the left term is zero since is an -FP-projective -module and the right term is zero since . Thus, , which implies that is an -FP-projective -module.
Let be a --pure submodule of an injective module . Then, is --absolutely pure by ([6], Theorem 3.2), and so by (4). Thus, the exactness of implies the injectivity of .
Let be a --absolutely pure -module. Then, is a --pure submodule of its injective envelope by ([6], Theorem 3.2). Hence, is injective by (5). Therefore, . □

Corollary 2. Every -FP-hereditary ring is a coherent ring.

Proof. This is a consequence of Proposition 11 and Proposition 9. □
The converse of Corollary 2 is not true in general (see, ([9], Example 3.9)).

Proposition 12. The following conditions are equivalent for a ring :(1)Every -module is -FP-projective(2) is -FP-projective for any ideal of(3)Every --absolutely pure -module is injective. If is composed of units, then the previous conditions are also equivalent to(4) is a Noetherian ring

Proof. This is trivial.
Let be a --absolutely pure -module. Then, for any ideal of , we have since is an -FP-projective -module. Hence, is an injective -module by ([12], Theorem 3.3.8).
Let be an -module. Then, for any --absolutely pure -module , we have since is an injective -module. Thus, is -FP-projective.
This follows by ([7], Proposition 2.6) since every -FP-projective -module is FP-projective.
Let be an -module. Then, is FP-projective since is Noetherian by ([7], Proposition 2.6) again. Hence, is -FP-projective since is composed of units.

Proposition 13. The following conditions are equivalent for a ring :(1)Every f.p. -module is -FP-projective(2)Every --absolutely pure -module is absolutely pure(3)Every FP-projective -module is -FP-projective

Proof. Let be a --absolutely pure -module. Then, for any f.p. -module , we have since is an -FP-projective -module. Hence, is an absolutely pure -module.
Let be an FP-projective -module. Then, for any --absolutely pure -module , we have since is an absolutely pure -module. Thus, is -FP-projective.
This follows from the fact that f.p. -modules are always FP-projective.

In the following proposition, we will prove that is -FP-projective if and only if and are -FP-projective. However, we need the following lemmas. For brevity’s sake, when are rings and (resp., ) is an -module (resp., -module), until the end of this section, we will sometimes set and .

Lemma 5. Let be an -module and be an -module and set . Then, is a --absolutely pure -module if and only if each is a --absolutely pure -module, .

Proof. Suppose that is a --absolutely pure -module and let be a finitely presented -module. It is clear that is also an -module (via the canonical projection ). With this modulation, and by using ([11], Theorem 2.1.8), is an f.p. -module. Thus, there exists such that . From ([14], Theorem 10.75), . Hence, is --torsion with respect to . Consequently, is a --absolutely pure -module. Similarly, is a --absolutely pure -module.
Conversely, assume that each is a --absolutely pure -module, , and let be an f.p. -module. Then, there exists and and set such that(by ([14], Theorem 10.74)).
On the other hand, by ([11], Theorem 2.1.8), (resp. ) is an f.p. -module (resp., -module). Thus, and , which imply that is --torsion with respect to and is --torsion with respect to . Consequently, is --torsion with respect to , and so is a --absolutely pure -module. □

Lemma 6. Let be a surjective ring homomorphism, where is projective as an -module. If is a --absolutely pure -module, then is a --absolutely pure -module.

Proof. Let be an f.p. -module. Then, there exists an exact sequence of -modules with f.g. and f.g. and projective. Since is a projective -module, we have the following exact sequence of -modules. Note that is an f.g. -module, and is an f.g. and projective -module. Thus, is an f.p. -module. Since is a --absolutely pure -module, there exists such that , and so is --torsion with respect to . Therefore, by ([14], Theorem 10.74), is --torsion with respect to some . Thus, is a --absolutely pure -module. □

Proposition 14. Let be an -module and let be a multiplicative subset of for , and set . Then, is an -FP-projective -module if and only if each is an -FP-projective -module for .

Proof. Suppose that is an -FP-projective -module, and let be a --absolutely pure -module. It is clear that is also an -module (via the canonical projection ). With this modulation, is a --absolutely pure -module by Lemma 6. Then, by reference [14], Theorem 10.74, we obtain the following isomorphisms:Consequently, is an -FP-projective -module. Similarly, is an -FP-projective -module.
Conversely, we assume that each is an -FP-projective -module for . Let be a --absolutely pure -module, and set for . It is clear that . By reference [14], Theorem 10.74, we have the following isomorphisms:On the other hand, by Lemma 5, (resp., ) is a --absolutely -module (resp., -module). Thus, and . Consequently, , and so is -FP-projective.

3. -FP-Projective Dimension of a Module and Global -FP-Projective Dimension of a Ring

This section introduces and investigates the -FP-projective dimension of a module and the global -FP-projective dimension of a ring.

Definition 3. Let be a ring. For any -module , the --projective dimension of , denoted by -, is the smallest integer such that for any --absolutely pure -module . If no such integer exists, we set -.
The global -FP-projective dimension of is defined as follows:Clearly, an -module is --projective if and only if and , with equality when consists of units. However, this inequality may be strict (Example 1). Also, with equality when consists of units, and this inequality may be strict. For example, consider a ring , the ring of integers. Since is Noetherian, we get (by ([7], Proposition 2.6)). Moreover, by Example 1, there exists an (FP-projective) -module which is not -FP-projective. Thus, .
First, we describe the -FP-projective dimension of a module over a coherent ring.

Proposition 15. Let be a coherent ring and be finite. The following conditions are equivalent for any -module :(1)(2) for any --absolutely pure -module (3) for any --absolutely pure -module and any (4)If a sequence is exact, where are -projective -modules, then is -projective(5)If a sequence is exact, where are projective -modules, then is -projective(6)There exists an exact sequence where each is -projective

Proof. These are clear.
Let be an exact sequence of -modules, where are -projective, and set and , where . Using Proposition 5, we get the following isomorphisms:for any --absolutely pure -module . Thus, is -projective.
These are obvious.
We proceed by induction on . For the case, is an -projective module, and so (3) holds by Proposition 5. If , then there is an exact sequence , where each is -projective. Set . Then, we have the following exact sequences:Hence, by induction, for any -module and all . Thus, . Hence, the desired result follows.

Proposition 16. Let be a coherent ring, be finite, and be an exact sequence of -modules. If two of , , and are finite, so is the third. Moreover, we have the following conditions:(1).(2).(3).

Proof. This follows from the standard of homological algebra.

Corollary 3. Let be a coherent ring, be finite, and be an exact sequence of -modules. If is --projective and -, then .

The proof of the following result is straightforward.

Proposition 17. Let be a coherent ring, be a finite multiplicative subset of , and be a family of -modules. Then, .

Proposition 18. Let be a ring and be an integer. Then, the following statements are equivalent:(1)(2) for any -module (3) for any ideal of (4) for any --absolutely pure -module Consequently, we have the following equalities:

Proof. Let be an -module. For every --absolutely pure -module , we have . Hence, .
These are clear.
Let be a --absolutely pure -module. For every ideal of , we have . Thus, . □
Next, we show that rings with are exactly Noetherian rings if is composed of units.

Proposition 19. Let be a ring and be composed of units. Then, the following conditions are equivalent:(1)(2)Every -module is --projective(3) is --projective for any ideal of(4)Every --absolutely pure -module is injective(5) is a Noetherian ring

Proof. The equivalence of (1), (2), (3), and (4) follow from Proposition 18.
This follows from Proposition 12.

Finally, we show that rings with are exactly -FP-hereditary rings.

Proposition 20. The following conditions are equivalent for a ring :(1)(2)Every submodule of --projective -module is --projective(3)Every submodule of projective -module is --projective(4) for any --absolutely pure -module (5) is an -FP-hereditary ring

Proof. These are obvious.
This follows by Proposition 18.
Let be a --absolutely pure -module and be an ideal of . The exact sequence gives rise to the following exact sequence:Thus, , and so .
Let be an ideal of . For any --absolutely pure -module , we have the following exact sequence:Thus, , which implies that is --projective. Therefore, is an -FP-hereditary ring.
This follows from Proposition 11. □

Proposition 21. Let be the product of rings and and a multiplicative subset of for each , and set . Then, is an -FP-hereditary ring if and only if is an -FP-hereditary ring for each .

Proof. This follows from Proposition 14 and Proposition 20. □
The following nontrivial example shows that FP-hereditary rings are not -FP-hereditary in general.

Example 2. Let , where is an FP-hereditary ring and is an FP-hereditary ring that is not hereditary (see ([9], Example 3.2(3)) for a concrete example for ). Then, is certainly FP-hereditary. We set . Then, is not -FP-hereditary by Proposition 1.9 since -FP-hereditary rings are exactly hereditary rings.

Data Availability

No underlying data were collected or produced in this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

H. Kim was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2021R1I1A3047469). X. Zhang was supported by the National Natural Science Foundation of China (No. 12061001).

References

X. L. Zhang, “Characterizing S-flat modules and S-von Neumann regular rings by uniformity,” Bulletin of the Korean Mathematical Society, vol. 59, no. 3, pp. 643–657, 2022.
View at: Google Scholar
X. L. Zhang, “The u-S-weak global dimensions of commutative rings,” Communications of the Korean Mathematical Society, vol. 38, no. 1, pp. 97–112, 2023.
View at: Google Scholar
B. H. Maddox, “Absolutely pure modules,” Proceedings of the American Mathematical Society, vol. 18, no. 1, pp. 155–158, 1967.
View at: Publisher Site | Google Scholar
C. Megibben, “Absolutely pure modules,” Proceedings of the American Mathematical Society, vol. 26, no. 4, pp. 561–566, 1970.
View at: Publisher Site | Google Scholar
B. Stenström, “Coherent rings and FP-injective modules,” Journal of the London Mathematical Society, no. 2, pp. 323–329, 1970.
View at: Publisher Site | Google Scholar
X. L. Zhang, “On uniformly-S-absolutely pure modules,” Journal of the Korean Mathematical Society.
View at: Publisher Site | Google Scholar
L. Mao and N. Ding, “FP-Projective dimensions,” Communications in Algebra, vol. 33, no. 4, pp. 1153–1170, 2005.
View at: Publisher Site | Google Scholar
A. Moradzadeh-Dehkordi and S. H. Shojaee, “Rings in which every ideal is pure-projective or FP-projective,” Journal of Algebra, vol. 478, pp. 419–436, 2017.
View at: Publisher Site | Google Scholar
R. A. K. Assaad, E. M. Bouba, and M. Tamekkante, “w-FP-projective modules and dimensions,” Algebra and Related Topics with Applications: ICARTA-2019, Springer Nature Singapore, Singapore, 2022.
View at: Google Scholar
L. Fuchs, Abelian Groups, Springer, Berlin, Germany, 2015.
S. Glaz, Commutative Coherent Rings, Springer, Berlin, Germany, 1989.
F. Wang and H. Kim, Foundations of Commutative Rings and Their Modules, Springer Nature Singapore Pte Ltd, Singapore, 2016.
F. Wang and H. Kim, “Two generalizations of projective modules and their applications,” Journal of Pure and Applied Algebra, vol. 219, no. 6, pp. 2099–2123, 2015.
View at: Publisher Site | Google Scholar
J. J. Rotman, An Introduction to Homological Algebra, Springer, New York, NY, USA, 2nd edition, 2009.

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Copyright © 2023 Refat Abdelmawla Khaled Assaad et al. 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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