#### 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* [1] 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 [9], 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 [9], 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* [2] or *absolutely pure* [3] 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* [4] 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 [5]. Following [5], 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 [6], using the two concepts and characterized right -coherent rings. Following [6], 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 [7], 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 [8]. Let be a torsion theory for the category of all right -modules; then, according to [8], 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 [4], 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 [9] if is closed under homomorphic images and extensions. Following [9], 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 [11], 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 -modules such 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*(4)

*X**for each -torsion-free injective module*

*E**Proof. * Use induction on . If , then the result holds by Proposition 2.5(3) in [4]. 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 [4]. 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 [10], 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 [4], and it shows that is -2-FP. In case, , then the result holds by Lemma 3.1 in [4]. 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 [4], 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 [6] 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 [4].

*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)

*(2)*

*R*is right -*n*-coherent*for any*(3)

*n*-presented right -module*A*and direct system of -torsion-free modules*for any*(4)

*n*-presented right -module*A*and each -torsion-free module*X**for any*(5)

*n*-presented right -module*A*and each -torsion-free injective module*E**If*(6)

*X*is a -*n*-FP-injective module, then*X*is*n*-FP-injective*Any direct limit of -torsion-free*(7)

*n*-FP-injective modules is*n*-FP-injective*Any direct limit of -torsion-free FP-injective modules is*(8)

*n*-FP-injective*Any direct limit of -torsion-free injective modules is*(9)

*n*-FP-injective*A -torsion-free module*(10)

*X*is*n*-FP-injective if and only if is*n*-flat*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 [10], 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 [10], the maps , , and are monic. Since is a hereditary torsion theory and is -torsion-free, by Proposition 3.2, Chap VI in [9], 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) [4], 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 [2]. Let be a direct system of -torsion-free -FP-injective modules. Then by Proposition 1 in [7], 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)

*(2)*

*R*is right*n*-coherent*for any*(3)

*n*-presented right -module*A*and direct system of right -modules*for any*(4)

*n*-presented right -module*A*and each right -module*X**for any*(5)

*n*-presented right -module*A*and each injective right -module*E**If*(6)

*X*is a weakly*n*-FP-injective module, then*X*is*n*-FP-injective*Any direct limit of*(7)

*n*-FP-injective modules is*n*-FP-injective*Any direct limit of FP-injective modules is*(8)

*n*-FP-injective*Any direct limit of injective modules is*(9)

*n*-FP-injective*A right -module*(10)

*X*is*n*-FP-injective if and only if is*n*-flat*If*

*Y*is a pure submodule of an*n*-FP-injective right -module*X*, then is*n*-FP-injectiveWe note that the equivalences of (1), (2), (6), and (9) in Corollary 2 appeared in Theorem 3.1 in [2].

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 [2]. Since is -FP-injective and is a pure submodule of , so, by Proposition 2.6 in [2], *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*(4)

*A*and each -torsion-free module*X**for any finitely presented right -module*(5)

*A*and each -torsion-free injective module*E**If*(6)

*X*is a -1-FP-injective module, then*X*is FP-injective*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*(9)

*X*is FP-injective if and only if is flat*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).