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Research Article | Open Access

Volume 2020 |Article ID 9651894 | https://doi.org/10.1155/2020/9651894

Zhu Zhanmin, "-Coherence Relative to a Hereditary Torsion Theory", Journal of Mathematics, vol. 2020, Article ID 9651894, 5 pages, 2020. https://doi.org/10.1155/2020/9651894

# -Coherence Relative to a Hereditary Torsion Theory

Accepted06 Jul 2020
Published04 Aug 2020

#### Abstract

Let be a ring, a hereditary torsion theory of mod-, and a positive integer. Then, is called right --coherent if every -presented right -module is -presented. We present some characterizations of right --coherent rings, as corollaries, and some characterizations of right -coherent rings and right -coherent rings are obtained.

#### 1. Introduction

Throughout this paper, is an associative ring with identity and all modules considered are unitary. For any -module , will be the character module of .

Recall that a torsion theory  for the category of all right -modules consists of two subclasses and such that(1) for all and (2)If for all , then (3)If for all , then

In this case, is called a torsion class and its objects are called -torsion, is called a torsion-free class, and its objects are called -torsion free. From, Proposition 2.1, Chap VI, in , a class of right -modules is a torsion class for some torsion theory if and only if is closed under quotient modules, direct sums, and extensions. From Proposition 2.2, Chap VI in , a class of right -modules is a torsion-free class for some torsion theory if and only if is closed under submodules, direct products, and extensions. A torsion theory is called hereditary if is closed under submodules.

We recall also that a right -module is called FP-injective  or absolutely pure  if for every finitely presented right -module ; a left -module is flat if and only if for every finitely presented right -module ; a ring is right coherent  if every finitely generated right ideal of is finitely presented, or equivalently, if every finitely generated submodule of a projective right -module is finitely presented. -injective modules, flat modules, coherent rings, and their generalizations have been studied extensively by many authors. For example, in 1994, Costa introduced the concept of right n-coherent rings in . Following , a ring is called right -coherent in case every -presented right -module is -presented, where a right -module is called -presented in case there exists an exact sequence of right -modules , in which every is finitely generated free. It is easy to see that a ring is right coherent if and only if is right 1-coherent. In 1996, Chen and Ding introduced the concepts of n-FP-injective modules and n-flat modules in , using the two concepts and characterized right -coherent rings. Following , a right -module is called --injective in case for every -presented right -module ; a left -module is called -flat in case for every -presented right -module .

Let be a (hereditary) torsion theory for the category of all right -modules. Then, according to , a right -module is called right -finitely generated (or -FG for short) if there exists a finitely generated submodule such that ; a right -module is called -finitely presented (or -FP for short) if there exists an exact sequence of right -modules with finitely generated free and -finitely generated; is called -coherent if every finitely generated right ideal of is -FP. In 1993, Nieves introduced the concept of -n-presented (or -n-FP for short) modules in . Let be a torsion theory for the category of all right -modules; then, according to , a right -module is called --presented in case there exists an exact sequence of right -modules , where each is finitely generated free and is -finitely generated. Let be a hereditary torsion theory. Then, by Theorem 3.3 in , it is easy to see that is right -coherent if and only if every finitely presented right -module is -2-FP.

In this article, we wish to extend the concepts of right -coherent rings and right -coherent rings to right --coherent rings and give some characterizations of these rings (see Theorems 1 and 2), as corollaries; some characterizations of right -coherent rings and right -coherent rings will be obtained (see Corollaries 2 and 4).

#### 2. Characterizations of --Presented Modules and Right --Coherent Rings

We recall that a nonempty subclass of right -modules is called a weak torsion class  if is closed under homomorphic images and extensions. Following , if a class of right -modules is a weak torsion class, then a right -module is called -finitely generated (or -FG for short) if there exists a finitely generated submodule such that ; a right -module is called -finitely presented (or -FP for short) if there exists an exact sequence of right -modules with finitely generated free and -finitely generated; a right -module is called -presented if there exists an exact sequence of right -modules:such that are finitely generated free and is -finitely generated, where is a positive integer. Let be a torsion theory of mod-R. Then, it is easy to see that is a weak torsion class. In this case, a right -module is -finitely generated if and only if it is -finitely generated, and a right -module is --presented if and only if it is -presented for any positive integer . Let be a right -module, a torsion theory of mod-R, and a positive integer. Then, form Proposition 3.4 in , the following conditions are equivalent:(1) is --presented.(2) is -presented, and if there exists an exact sequence of right -modules:such that are finitely generated free; then, is -finitely generated.(3)There exists an exact sequence of right -modules:such that is finitely generated free and is --presented.If , then the above conditions are also equivalent to(4) is -presented, and if there exists an exact sequence of right -modulessuch that are finitely generated free; then, is -finitely presented.

Now, we give the characterizations of --presented modules.

Theorem 1. Let be a torsion theory of mod-R, a nonnegative integer, and an -presented right -module. Then, the following statements are equivalent for A:(1) is --presented(2)The canonical map is an isomorphism for each direct system of -torsion-free modules(3) for each -torsion-free module X(4) for each -torsion-free injective module E

Proof. Use induction on . If , then the result holds by Proposition 2.5(3) in . Assume that the result holds when . Then, when , suppose is a --presented module. Let be an exact sequence of right -modules, where is finitely generated free and is --presented. Then, we have a commutative diagram:with exact rows. Since is an isomorphism and hence epic and is an isomorphism by hypothesis, we have that is also an isomorphism by the Five Lemma. Use induction on . If , then the result holds by Proposition 2.5 (3) in . Assume that the result holds when . Then, when , suppose is a -presented right -module. Let be an exact sequence of right -modules, where is finitely generated free and is -presented. Then, for any direct system of -torsion-free modules. If , then we have a commutative diagram:with exact rows. Since is an isomorphism by condition, we have that is also an isomorphism. So, by hypothesis, is --presented, and hence is --presented. If , then we have a commutative diagram:with exact rows. From 25.4 (d) in , is an isomorphism and hence epic, and is an isomorphism by condition. Note that is an isomorphism, so, by the Five Lemma, we have that is also an isomorphism. So, is -FP by Proposition 2.5 in , and it shows that is -2-FP. In case, , then the result holds by Lemma 3.1 in . In case, , then there is an exact sequence of right -modules , where is finitely generated free and is -FP. And then we have a commutative diagram:with exact rows. By Lemma 3.1 , and are isomorphisms. So, by the Five Lemma, is also an isomorphism. In case, , then we have an exact sequence of right -modules , where each is finitely generated free and is -2-FP, and hence we have , as required. It is obvious. Since is -FP, there exists an exact sequence of right -modules , where each is finitely generated free and is finitely presented. Thus, for any -torsion-free injective module by (4). It follows from Proposition 2 in  that is -2-FP, and therefore is --presented.

Let be a right -module and be a positive integer. If , then it is easy to see that is -presented if and only if it is -presented. If , then it is easy to see that is -presented if and only if it is -presented.

Corollary 1. Let be a nonnegative integer and an -presented right -module. Then, the following statements are equivalent:(1) is -presented(2)The canonical map is an isomorphism for each direct system of right R-modules(3) for each right -module X(4) for each injective right -module E

Definition 1. Let be a torsion theory of mod-R. Then, the ring is called right -n-coherent, if every n-presented right R-module is -presented.
Let be a torsion theory of mod-R. Then, it is easy to see that is right --coherent if and only if every -presented right -module is -presented.

Example 1. (1)Let . Then, is right --coherent if and only if is right -coherent.(2) is right -coherent if and only if is right -1-coherent.(3)Let . Then, is right --coherent.

Proof. (1) and (3) are obvious. (2) follows from Theorem 3.3 (2) in .

Definition 2. Let be a torsion theory of mod-R and n a positive integer. Then, a right -module M is said to be -n-FP-injective, if for each --presented module M; a right -module M is said to be -FP-injective if it is -1-FP-injective.
Clearly, each -FP-injective module is --FP-injective. If , then it is easy to see that a right -module is --FP-injective if and only if it is -FP-injective. Now, we give our characterization of right --coherent rings.

Theorem 2. Let be a hereditary torsion theory of mod-R and n a positive integer. Then, the following statements are equivalent for the ring R:(1)R is right -n-coherent(2) for any n-presented right -module A and direct system of -torsion-free modules(3) for any n-presented right -module A and each -torsion-free module X(4) for any n-presented right -module A and each -torsion-free injective module E(5)If X is a -n-FP-injective module, then X is n-FP-injective(6)Any direct limit of -torsion-free n-FP-injective modules is n-FP-injective(7)Any direct limit of -torsion-free FP-injective modules is n-FP-injective(8)Any direct limit of -torsion-free injective modules is n-FP-injective(9)A -torsion-free module X is n-FP-injective if and only if is n-flat(10)If Y is a pure submodule of a -torsion-free n-FP-injective module X, then is n-FP-injective.

Proof. follows from Theorem 1., , and are obvious. Let , where each is a -torsion-free -FP-injective module. Then, for any --FP module , by Theorem 1, we have that , so is --FP-injective and thus it is -FP-injective by (5).. Let be an -presented right -module with a finite -presentation . Write and . Then, is finitely generated, and we get an exact sequence of right -modules . Let be any direct system of -torsion-free injective right -modules (with directed). Then, is -FP-injective by (8), so and hence . Thus, we have a commutative diagram:with exact rows. Since and are isomorphisms by 25.4(d) in , is an isomorphism by the Five Lemma. Now, let be any direct system of -torsion-free modules (with directed). Then, we have a commutative diagram with exact rows:where is the injective hull of . Since is finitely generated, by 24.9 in , the maps , , and are monic. Since is a hereditary torsion theory and is -torsion-free, by Proposition 3.2, Chap VI in , is -torsion-free. And so, by the above proof, is an isomorphism. Hence, is also an isomorphism by the Five Lemma again, and then is -finitely presented by Proposition 2.5 (3) , and thus is --presented. Therefore, is right --coherent. Since is a pure submodule, the pure exact sequence induces a split exact sequence . Since is -torsion-free and -FP-injective, by (9), is -flat, so is also -flat, and thus is -FP-injective by Corollary 2.8 in . Let be a direct system of -torsion-free -FP-injective modules. Then by Proposition 1 in , we have a map-pure, and hence pure exact sequence . Observing that is -torsion-free and -FP-injective, by (10), we have that is -FP-injective.

We call a right -module weakly n-FP-injective if for any -presented right -module . Let . Then, we have the following results.

Corollary 2. Let n be a positive integer. Then, the following statements are equivalent for a ring R:(1)R is right n-coherent(2) for any n-presented right -module A and direct system of right -modules(3) for any n-presented right -module A and each right -module X(4) for any n-presented right -module A and each injective right -module E(5)If X is a weakly n-FP-injective module, then X is n-FP-injective(6)Any direct limit of n-FP-injective modules is n-FP-injective(7)Any direct limit of FP-injective modules is n-FP-injective(8)Any direct limit of injective modules is n-FP-injective(9)A right -module X is n-FP-injective if and only if is n-flat(10)If Y is a pure submodule of an n-FP-injective right -module X, then is n-FP-injective

We note that the equivalences of (1), (2), (6), and (9) in Corollary 2 appeared in Theorem 3.1 in .

Corollary 3. Let be a hereditary torsion theory of mod-R and n a positive integer. If R is right -n-coherent, then a -torsion-free module X is n-FP-injective if and only if is n-FP-injective.

Proof. Let be a -torsion-free -FP-injective module. Since is right --coherent, by Theorem 2 (9), is -flat, and so is -FP-injective by Proposition 2.3 in . Since is -FP-injective and is a pure submodule of , so, by Proposition 2.6 in , X is -FP-injective.

Let ; then, by Theorem 2, we can obtained several results on right -coherent rings.

Corollary 4. Let be a hereditary torsion theory of mod-R. Then, the following statements are equivalent for the ring R:(1)R is right -coherent(2) for any finitely presented right -module A and direct system of -torsion-free modules(3) for any finitely presented right -module A and each -torsion-free module X(4) for any finitely presented right -module A and each -torsion-free injective module E(5)If X is a -1-FP-injective module, then X is FP-injective(6)Any direct limit of -torsion-free FP-injective modules is FP-injective(7)Any direct limit of -torsion-free injective modules is FP-injective.(8)A -torsion-free module X is FP-injective if and only if is flat(9)If Y is a pure submodule of a -torsion-free FP-injective module X, then is FP-injective

#### Data Availability

No data were used to support this study.

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.

#### Acknowledgments

This research was supported by the Natural Science Foundation of Zhejiang Province, China (LY18A010018).

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