Let R be a ring, τ=T,ℱ a hereditary torsion theory of mod-R, and n a positive integer. Then, R is called right τ-n-coherent if every n-presented right R-module is τ,n+1-presented. We present some characterizations of right τ-n-coherent rings, as corollaries, and some characterizations of right n-coherent rings and right τ-coherent rings are obtained.

Natural Science Foundation of Zhejiang ProvinceLY18A0100181. Introduction

Throughout this paper, R is an associative ring with identity and all modules considered are unitary. For any R-module M, M+=HomM,ℚ/ℤ will be the character module of M.

Recall that a torsion theory [1] τ=T,ℱ for the category of all right R-modules consists of two subclasses T and ℱ such that

HomT,F=0 for all T∈T and F∈ℱ

If HomT,F=0 for all F∈ℱ, then T∈T

If HomT,F=0 for all T∈T, then F∈ℱ

In this case, T is called a torsion class and its objects are called τ-torsion, ℱ is called a torsion-free class, and its objects are called τ-torsion free. From, Proposition 2.1, Chap VI, in [9], a class T of right R-modules is a torsion class for some torsion theory if and only if T is closed under quotient modules, direct sums, and extensions. From Proposition 2.2, Chap VI in [9], a class ℱ of right R-modules is a torsion-free class for some torsion theory if and only if ℱ is closed under submodules, direct products, and extensions. A torsion theory τ=T,ℱ is called hereditary if T is closed under submodules.

We recall also that a right R-module M is called FP-injective [2] or absolutely pure [3] if ExtR1A,M=0 for every finitely presented right R-module A; a left R-module M is flat if and only if Tor1RA,M=0 for every finitely presented right R-module A; a ring R is right coherent [4] if every finitely generated right ideal of R is finitely presented, or equivalently, if every finitely generated submodule of a projective right R-module is finitely presented. FP-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 R is called right n-coherent in case every n-presented right R-module is n+1-presented, where a right R-module A is called n-presented in case there exists an exact sequence of right R-modules Fn⟶Fn−1⟶⋯⟶F1⟶F0⟶A⟶0, in which every Fi is finitely generated free. It is easy to see that a ring R is right coherent if and only if R 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 n-coherent rings. Following [6], a right R-module M is called n-FP-injective in case ExtRnA,M=0 for every n-presented right R-module A; a left R-module M is called n-flat in case TornRA,M=0 for every n-presented right R-module A.

Let τ=T,ℱ be a (hereditary) torsion theory for the category of all right R-modules. Then, according to [7], a right R-module M is called right τ-finitely generated (or τ-FG for short) if there exists a finitely generated submodule N such that M/N∈T; a right R-module A is called τ-finitely presented (or τ-FP for short) if there exists an exact sequence of right R-modules 0⟶K⟶F⟶A⟶0 with F finitely generated free and Kτ-finitely generated; R is called τ-coherent if every finitely generated right ideal of R is τ-FP. In 1993, Nieves introduced the concept of τ-n-presented (or τ-n-FP for short) modules in [8]. Let τ=T,ℱ be a torsion theory for the category of all right R-modules; then, according to [8], a right R-module A is called τ-n-presented in case there exists an exact sequence of right R-modules 0⟶Kn−1⟶Fn−1⟶⋯⟶F1⟶F0⟶A⟶0, where each Fi is finitely generated free and Kn−1 is τ-finitely generated. Let τ=T,ℱ be a hereditary torsion theory. Then, by Theorem 3.3 in [4], it is easy to see that R is right τ-coherent if and only if every finitely presented right R-module is τ-2-FP.

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

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We recall that a nonempty subclass T of right R-modules is called a weak torsion class [9] if T is closed under homomorphic images and extensions. Following [9], if a class T of right R-modules is a weak torsion class, then a right R-module M is called T-finitely generated (or T-FG for short) if there exists a finitely generated submodule N such that M/N∈T; a right R-module A is called T-finitely presented (or T-FP for short) if there exists an exact sequence of right R-modules 0⟶K⟶F⟶A⟶0 with F finitely generated free and KT-finitely generated; a right R-module A is called T,n-presented if there exists an exact sequence of right R-modules:(1)0⟶Kn−1⟶Fn−1⟶⋯⟶F1⟶F0⟶M⟶0,such that F0,⋯,Fn−1 are finitely generated free and Kn−1 is T-finitely generated, where n is a positive integer. Let τ=T,ℱ be a torsion theory of mod-R. Then, it is easy to see that T is a weak torsion class. In this case, a right R-module A is τ-finitely generated if and only if it is T-finitely generated, and a right R-module A is τ-n-presented if and only if it is T,n-presented for any positive integer n. Let A be a right R-module, τ a torsion theory of mod-R, and n a positive integer. Then, form Proposition 3.4 in [11], the following conditions are equivalent:

A is τ-n-presented.

A is n−1-presented, and if there exists an exact sequence of right R-modules:(2)0⟶Kn−1⟶Fn−1⟶⋯⟶F1⟶F0⟶A⟶0,

such that F0,⋯,Fn−1 are finitely generated free; then, Kn−1 is τ-finitely generated.

There exists an exact sequence of right R-modules:(3)0⟶K⟶F⟶A⟶0,

such that F is finitely generated free and K is τ-n−1-presented.

If n≥2, then the above conditions are also equivalent to

A is n−2-presented, and if there exists an exact sequence of right R-modules(4)0⟶Kn−2⟶Fn−2⟶⋯⟶F1⟶F0⟶A⟶0,

such that F0,⋯,Fn−2 are finitely generated free; then, Kn−2 is τ-finitely presented.

Now, we give the characterizations of τ-n+1-presented modules.

Theorem 1.

Let τ be a torsion theory of mod-R, n a nonnegative integer, and A an n-presented right R-module. Then, the following statements are equivalent for A:

A is τ-n+1-presented

The canonical map lim⟶ExtRnA,Xi⟶ExtRnA,lim⟶Xi is an isomorphism for each direct system Xii∈I of τ-torsion-free modules

TornRA,X+≅ExtRnA,X+ for each τ-torsion-free module X

TornRA,E+=0 for each τ-torsion-free injective module E

Proof.

1⇒2 Use induction on n. If n=0, then the result holds by Proposition 2.5(3) in [4]. Assume that the result holds when n=k. Then, when n=k+1, suppose A is a τ-k+2-presented module. Let 0⟶N⟶F⟶A⟶0 be an exact sequence of right R-modules, where F is finitely generated free and N is τ-k+1-presented. Then, we have a commutative diagram:(5)

with exact rows. Since ϕ1 is an isomorphism and hence epic and ϕ2 is an isomorphism by hypothesis, we have that ϕ3 is also an isomorphism by the Five Lemma.

2⇒1 Use induction on n. If n=0, then the result holds by Proposition 2.5 (3) in [4]. Assume that the result holds when n=k. Then, when n=k+1, suppose A is a k+1-presented right R-module. Let 0⟶N⟶F⟶A⟶0 be an exact sequence of right R-modules, where F is finitely generated free and N is k-presented. Then, for any direct system Xii∈I of τ-torsion-free modules. If k>0, then we have a commutative diagram:(6)

with exact rows. Since ϕ3 is an isomorphism by condition, we have that ϕ2 is also an isomorphism. So, by hypothesis, N is τ-k+1-presented, and hence A is τ-k+2-presented. If k=0, then we have a commutative diagram:(7)

with exact rows. From 25.4 (d) in [10], ϕ0 is an isomorphism and hence epic, and ϕ3 is an isomorphism by condition. Note that ϕ1 is an isomorphism, so, by the Five Lemma, we have that ϕ2 is also an isomorphism. So, N is τ-FP by Proposition 2.5 in [4], and it shows that A is τ-2-FP.

1⇒3 In case, n=0, then the result holds by Lemma 3.1 in [4]. In case, n=1, then there is an exact sequence of right R-modules 0⟶K⟶F⟶A⟶0, where F is finitely generated free and K is τ-FP. And then we have a commutative diagram:(8)

with exact rows. By Lemma 3.1 [4], g and h are isomorphisms. So, by the Five Lemma, f is also an isomorphism. In case, n>1, then we have an exact sequence of right R-modules 0⟶Kn−2⟶Fn−2⟶⋯⟶F1⟶F0⟶M⟶0, where each Fi is finitely generated free and Kn−2 is τ-2-FP, and hence we have TornRA,X+≅Tor1RKn−2,X+≅ExtR1Kn−2,X+≅ExtRnA,X+, as required.

3⇒4 It is obvious.

4⇒1 Since A is n-FP, there exists an exact sequence of right R-modules 0⟶Kn−2⟶Fn−2⟶⋯⟶F1⟶F0⟶M⟶0, where each Fi is finitely generated free and Kn−2 is finitely presented. Thus, Tor1RKn−2,E+≅TornRA,E+=0 for any τ-torsion-free injective module E by (4). It follows from Proposition 2 in [6] that Kn−2 is τ-2-FP, and therefore A is τ-n+1-presented.

Let M be a right R-module and n be a positive integer. If τ=0,mod−R, then it is easy to see that M is τ,n+1-presented if and only if it is n+1-presented. If τ=mod−R,0, then it is easy to see that M is τ,n+1-presented if and only if it is n-presented.

Corollary 1.

Let n be a nonnegative integer and A an n-presented right R-module. Then, the following statements are equivalent:

A is n+1-presented

The canonical map lim⟶ExtRnA,Xi⟶ExtRnA,lim⟶Xi is an isomorphism for each direct system Xii∈I of right R-modules

TornRA,X+≅ExtRnA,X+ for each right R-module X

TornRA,E+=0 for each injective right R-module E

Definition 1.

Let τ be a torsion theory of mod-R. Then, the ring R is called right τ-n-coherent, if every n-presented right R-module is τ,n+1-presented.

Let τ=T,ℱ be a torsion theory of mod-R. Then, it is easy to see that R is right τ-n-coherent if and only if every n-presented right R-module is T,n+1-presented.

Example 1.

Let τ=0,mod−R. Then, R is right τ-n-coherent if and only if R is right n-coherent.

R is right τ-coherent if and only if R is right τ-1-coherent.

Let τ=mod−R,0. Then, R is right τ-n-coherent.

Proof.

(1) and (3) are obvious. (2) follows from Theorem 3.3 (2) in [4].

Definition 2.

Let τ be a torsion theory of mod-R and n a positive integer. Then, a right R-module M is said to be τ-n-FP-injective, if ExtRnA,M=0 for each τ-n+1-presented module M; a right R-module M is said to be τ-FP-injective if it is τ-1-FP-injective.

Clearly, each n-FP-injective module is τ-n-FP-injective. If τ=mod−R,0, then it is easy to see that a right R-module M is τ-n-FP-injective if and only if it is n-FP-injective. Now, we give our characterization of right τ-n-coherent rings.

Theorem 2.

Let τ be a hereditary torsion theory of mod-R and n a positive integer. Then, the following statements are equivalent for the ring R:

R is right τ-n-coherent

lim⟶ExtRnA,Xi≅ExtRnA,lim⟶Xi for any n-presented right R-module A and direct system Xii∈I of τ-torsion-free modules

TornRA,X+≅ExtRnA,X+ for any n-presented right R-module A and each τ-torsion-free module X

TornRA,E+=0 for any n-presented right R-module A and each τ-torsion-free injective module E

If X is a τ-n-FP-injective module, then X is n-FP-injective

Any direct limit of τ-torsion-free n-FP-injective modules is n-FP-injective

Any direct limit of τ-torsion-free FP-injective modules is n-FP-injective

Any direct limit of τ-torsion-free injective modules is n-FP-injective

A τ-torsion-free module X is n-FP-injective if and only if X+ is n-flat

If Y is a pure submodule of a τ-torsion-free n-FP-injective module X, then X/Y is n-FP-injective.

Proof.

1⇔2⇔3⇔4 follows from Theorem 1.

1⇒5, 6⇒7⇒8, and 3⇒9 are obvious.

5⇒6 Let X=lim⟶Xi, where each Xii∈I is a τ-torsion-free n-FP-injective module. Then, for any τ-n+1-FP module A, by Theorem 1, we have that ExtRnA,X=ExtRnA,lim⟶Xi≅lim⟶ExtRnA,Xi=0, so X is τ-n-FP-injective and thus it is n-FP-injective by (5).

8⇒1. Let A be an n-presented right R-module with a finite n-presentation Fn⟶dnFn−1⟶dn−1⋯⟶F2⟶d2F1⟶d1F0⟶d0A⟶0. Write Kn−1=Kerdn−1 and Kn−2=Kerdn−2. Then, Kn−1 is finitely generated, and we get an exact sequence of right R-modules 0⟶Kn−1⟶Fn−1⟶Kn−2⟶0. Let Eii∈I be any direct system of τ-torsion-free injective right R-modules (with I directed). Then, lim⟶Ei is n-FP-injective by (8), so ExtRnA,lim⟶Ei=0 and hence ExtR1Kn−2,lim⟶Ei=0. Thus, we have a commutative diagram:(9)

with exact rows. Since f and g are isomorphisms by 25.4(d) in [10], h is an isomorphism by the Five Lemma. Now, let Xii∈I be any direct system of τ-torsion-free modules (with I directed). Then, we have a commutative diagram with exact rows:(10)

where EXi is the injective hull of Xi. Since Kn−1 is finitely generated, by 24.9 in [10], the maps ϕ1, ϕ2, and ϕ3 are monic. Since τ is a hereditary torsion theory and Xi is τ-torsion-free, by Proposition 3.2, Chap VI in [9], EXi is τ-torsion-free. And so, by the above proof, ϕ2 is an isomorphism. Hence, ϕ1 is also an isomorphism by the Five Lemma again, and then Kn−1 is τ-finitely presented by Proposition 2.5 (3) [4], and thus A is τ-n+1-presented. Therefore, R is right τ-n-coherent.

9⇒10 Since Y is a pure submodule, the pure exact sequence 0⟶Y⟶X⟶X/Y⟶0 induces a split exact sequence 0⟶X/Y+⟶X+⟶Y+⟶0. Since X is τ-torsion-free and n-FP-injective, by (9), X+ is n-flat, so X/Y+ is also n-flat, and thus X/Y is n-FP-injective by Corollary 2.8 in [2].

10⇒6 Let Xii∈I be a direct system of τ-torsion-free n-FP-injective modules. Then by Proposition 1 in [7], we have a map-pure, and hence pure exact sequence 0⟶K⟶⊕i∈IXi⟶lim⟶Xi⟶0. Observing that ⊕i∈IXi is τ-torsion-free and n-FP-injective, by (10), we have that lim⟶Xi is n-FP-injective.

We call a right R-module Xweakly n-FP-injective if ExtRnA,X=0 for any n+1-presented right R-module A. Let τ=0,mod−R. Then, we have the following results.

Corollary 2.

Let n be a positive integer. Then, the following statements are equivalent for a ring R:

R is right n-coherent

lim⟶ExtRnA,Xi≅ExtRnA,lim⟶Xi for any n-presented right R-module A and direct system Xii∈I of right R-modules

TornRA,X+≅ExtRnA,X+ for any n-presented right R-module A and each right R-module X

TornRA,E+=0 for any n-presented right R-module A and each injective right R-module E

If X is a weakly n-FP-injective module, then X is n-FP-injective

Any direct limit of n-FP-injective modules is n-FP-injective

Any direct limit of FP-injective modules is n-FP-injective

Any direct limit of injective modules is n-FP-injective

A right R-module X is n-FP-injective if and only if X+ is n-flat

If Y is a pure submodule of an n-FP-injective right R-module X, then X/Y is n-FP-injective

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

Corollary 3.

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

Proof.

⇒ Let X be a τ-torsion-free n-FP-injective module. Since R is right τ-n-coherent, by Theorem 2 (9), X+ is n-flat, and so X++ is n-FP-injective by Proposition 2.3 in [2].

⇐ Since X++ is n-FP-injective and X is a pure submodule of X++, so, by Proposition 2.6 in [2], X is n-FP-injective.

Let n=1; then, by Theorem 2, we can obtained several results on right τ-coherent rings.

Corollary 4.

Let τ be a hereditary torsion theory of mod-R. Then, the following statements are equivalent for the ring R:

R is right τ-coherent

lim⟶ExtR1A,Xi≅ExtR1A,lim⟶Xi for any finitely presented right R-module A and direct system Xii∈I of τ-torsion-free modules

Tor1RA,X+≅ExtRnA,X+ for any finitely presented right R-module A and each τ-torsion-free module X

Tor1RA,E+=0 for any finitely presented right R-module A and each τ-torsion-free injective module E

If X is a τ-1-FP-injective module, then X is FP-injective

Any direct limit of τ-torsion-free FP-injective modules is FP-injective

Any direct limit of τ-torsion-free injective modules is FP-injective.

A τ-torsion-free module X is FP-injective if and only if X+ is flat

If Y is a pure submodule of a τ-torsion-free FP-injective module X, then X/Y 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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