# n-Coherence Relative to a Hereditary Torsion Theory

Authors: Zhu Zhanmin
Journal: Journal of Mathematics
 (2020)

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## Abstract

LetR 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.


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## Body

## 1. 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 atorsion theory [1] τ=T,ℱ for the category of all right R-modules consists of two subclasses T and ℱ such that(1)
HomT,F=0 for all T∈T and F∈ℱ(2)
IfHomT,F=0 for all F∈ℱ, then T∈T(3)
IfHomT,F=0 for all T∈T, then F∈ℱIn 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 rightR-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.Letτ=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 rightτ-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).


## 2. Characterizations ofτ-n+1-Presented Modules and Right τ-n-Coherent Rings

We recall that a nonempty subclassT 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:(1)
A is τ-n-presented.(2)
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 thatF0,⋯,Fn−1 are finitely generated free; then, Kn−1 is τ-finitely generated.(3)
There exists an exact sequence of rightR-modules:(3)0⟶K⟶F⟶A⟶0,such thatF is finitely generated free and K is τ-n−1-presented.Ifn≥2, then the above conditions are also equivalent to(4)
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 thatF0,⋯,Fn−2 are finitely generated free; then, Kn−2 is τ-finitely presented.Now, we give the characterizations ofτ-n+1-presented modules.Theorem 1.
Letτ 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:(1)
A is τ-n+1-presented(2)
The canonical maplim⟶ExtRnA,Xi⟶ExtRnA,lim⟶Xi is an isomorphism for each direct system Xii∈I of τ-torsion-free modules(3)
TornRA,X+≅ExtRnA,X+ for each τ-torsion-free module X(4)
TornRA,E+=0 for each τ-torsion-free injective module EProof.
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. Sinceϕ1 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. Sinceϕ3 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.LetM 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.
Letn be a nonnegative integer and A an n-presented right R-module. Then, the following statements are equivalent:(1)
A is n+1-presented(2)
The canonical maplim⟶ExtRnA,Xi⟶ExtRnA,lim⟶Xi is an isomorphism for each direct system Xii∈I of right R-modules(3)
TornRA,X+≅ExtRnA,X+ for each right R-module X(4)
TornRA,E+=0 for each injective right R-module EDefinition 1.
Letτ 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.
Letτ=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.
(1)
Letτ=0,mod−R. Then, R is right τ-n-coherent if and only if R is right n-coherent.(2)
R is right τ-coherent if and only if R is right τ-1-coherent.(3)
Letτ=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.
Letτ 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, eachn-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.
Letτ 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 τ-n-coherent(2)
lim⟶ExtRnA,Xi≅ExtRnA,lim⟶Xi for any n-presented right R-module A and direct system Xii∈I of τ-torsion-free modules(3)
TornRA,X+≅ExtRnA,X+ for any n-presented right R-module A and each τ-torsion-free module X(4)
TornRA,E+=0 for any n-presented right R-module A and each τ-torsion-free injective module E(5)
IfX is a τ-n-FP-injective module, then X is n-FP-injective(6)
Any direct limit ofτ-torsion-free n-FP-injective modules is n-FP-injective(7)
Any direct limit ofτ-torsion-free FP-injective modules is n-FP-injective(8)
Any direct limit ofτ-torsion-free injective modules is n-FP-injective(9)
Aτ-torsion-free module X is n-FP-injective if and only if X+ is n-flat(10)
IfY 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. Sincef 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)whereEXi 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 rightR-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.
Letn be a positive integer. Then, the following statements are equivalent for a ring R:(1)
R is right n-coherent(2)
lim⟶ExtRnA,Xi≅ExtRnA,lim⟶Xi for any n-presented right R-module A and direct system Xii∈I of right R-modules(3)
TornRA,X+≅ExtRnA,X+ for any n-presented right R-module A and each right R-module X(4)
TornRA,E+=0 for any n-presented right R-module A and each injective right R-module E(5)
IfX is a weakly n-FP-injective module, then X is n-FP-injective(6)
Any direct limit ofn-FP-injective modules is n-FP-injective(7)
Any direct limit of FP-injective modules isn-FP-injective(8)
Any direct limit of injective modules isn-FP-injective(9)
A rightR-module X is n-FP-injective if and only if X+ is n-flat(10)
IfY is a pure submodule of an n-FP-injective right R-module X, then X/Y is n-FP-injectiveWe note that the equivalences of (1), (2), (6), and (9) in Corollary2 appeared in Theorem 3.1 in [2].Corollary 3.
Letτ 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.Letn=1; then, by Theorem 2, we can obtained several results on right τ-coherent rings.Corollary 4.
Letτ be a hereditary torsion theory of mod-R. Then, the following statements are equivalent for the ring R:(1)
R is right τ-coherent(2)
lim⟶ExtR1A,Xi≅ExtR1A,lim⟶Xi for any finitely presented right R-module A and direct system Xii∈I of τ-torsion-free modules(3)
Tor1RA,X+≅ExtRnA,X+ for any finitely presented right R-module A and each τ-torsion-free module X(4)
Tor1RA,E+=0 for any finitely presented right R-module A and each τ-torsion-free injective module E(5)
IfX is a τ-1-FP-injective module, then X is FP-injective(6)
Any direct limit ofτ-torsion-free FP-injective modules is FP-injective(7)
Any direct limit ofτ-torsion-free injective modules is FP-injective.(8)
Aτ-torsion-free module X is FP-injective if and only if X+ is flat(9)
IfY is a pure submodule of a τ-torsion-free FP-injective module X, then X/Y is FP-injective



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