n-Coherence Relative to a Hereditary Torsion Theory

<jats:p>Let <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M2"><mml:mi>R</mml:mi></mml:math> be a ring, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M3"><mml:mi>τ</mml:mi><mml:mo>=</mml:mo><mml:mfenced open="(" close=")" separators="|"><mml:mrow><mml:mi mathvariant="script">T</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">ℱ</mml:mi></mml:mrow></mml:mfenced></mml:math> a hereditary torsion theory of mod-<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M4"><mml:mi>R</mml:mi></mml:math>, and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M5"><mml:mi>n</mml:mi></mml:math> a positive integer. Then, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M6"><mml:mi>R</mml:mi></mml:math> is called right <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M7"><mml:mi>τ</mml:mi></mml:math>-<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M8"><mml:mi>n</mml:mi></mml:math>-coherent if every <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M9"><mml:mi>n</mml:mi></mml:math>-presented right <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M10"><mml:mi>R</mml:mi></mml:math>-module is <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M11"><mml:mfenced open="(" close=")" separators="|"><mml:mrow><mml:mi>τ</mml:mi><mml:mo>,</mml:mo><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:mfenced></mml:math>-presented. We present some characterizations of right <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M12"><mml:mi>τ</mml:mi></mml:math>-<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M13"><mml:mi>n</mml:mi></mml:math>-coherent rings, as corollaries, and some characterizations of right <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M14"><mml:mi>n</mml:mi></mml:math>-coherent rings and right <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M15"><mml:mi>τ</mml:mi></mml:math>-coherent rings are obtained.</jats:p>


Introduction
roughout this paper, R is an associative ring with identity and all modules considered are unitary. For any R-module M, M + � Hom(M, (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 (1) Hom(T, F) � 0 for all T ∈ T and F ∈ F (2) If Hom(T, F) � 0 for all F ∈ F, then T ∈ T (3) If Hom(T, 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 Ext 1 R (A, M) � 0 for every finitely presented right R-module A; a left R-module M is flat if and only if Tor R 1 (A, 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 F n ⟶ F n−1 ⟶ · · · ⟶ F 1 ⟶ F 0 ⟶ A ⟶ 0, in which every F i 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 Let τ � (T, F) be a (hereditary) torsion theory for the category of all right R-modules. en, 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 K τ-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 ⟶ K n−1 ⟶ F n−1 ⟶ · · · ⟶ F 1 ⟶ F 0 ⟶ A ⟶ 0, where each F i is finitely generated free and K n−1 is τ-finitely generated. Let τ � (T, F) be a hereditary torsion theory. en, by eorem 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 eorems 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).

Modules and Right τ-n-Coherent Rings
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 K T-finitely generated; a right R-module A is called (T, n)-presented if there exists an exact sequence of right R-modules: such that F 0 , · · · , F n−1 are finitely generated free and K n−1 is T-finitely generated, where n is a positive integer. Let τ � (T, F) be a torsion theory of mod-R. en, 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. en, form Proposition 3.4 in [11], the following conditions are equivalent: (2) A is (n − 1)-presented, and if there exists an exact sequence of right R-modules: such that F 0 , · · · , F n−1 are finitely generated free; then, K n−1 is τ-finitely generated. (3) ere exists an exact sequence of right R-modules: 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 (4) A is (n − 2)-presented, and if there exists an exact sequence of right R-modules such that F 0 , · · · , F n−2 are finitely generated free; then, K n−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. en, the following statements are equivalent for A: (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. en, 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. en, we have a commutative diagram: with exact rows. Since ϕ 1 is an isomorphism and hence epic and ϕ 2 is an isomorphism by hypothesis, we have that ϕ 3 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. en, 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. en, for any direct system X i i∈I of τ-torsion-free modules. If k > 0, then we have a commutative diagram: 2 Journal of Mathematics with exact rows. Since ϕ 3 is an isomorphism by condition, we have that ϕ 2 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: with exact rows. From 25.4 (d) in [10], ϕ 0 is an isomorphism and hence epic, and ϕ 3 is an isomorphism by condition. Note that ϕ 1 is an isomorphism, so, by the Five Lemma, we have that ϕ 2 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 where F is finitely generated free and K is τ-FP. And then we have a commutative diagram: 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 ⟶ K n−2 ⟶ F n−2 ⟶ · · · ⟶ F 1 ⟶ F 0 ⟶ M ⟶ 0, where each F i is finitely generated free and K n−2 is τ-2-FP, and hence we have where each F i is finitely generated free and K n−2 is finitely presented. us, Tor R 1 (K n−2 , E + ) � Tor R n (A, E + ) � 0 for any τ-torsionfree injective module E by (4). It follows from Proposition 2 in [6] that K n−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. en, the following statements are equivalent: Let τ be a torsion theory of mod-R. en, 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. en, 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.
Proof. (1) and (3)   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-FPinjective. 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. en, the following statements are equivalent for the ring R:
(1) R is right τ-n-coherent for any n-presented right R-module A and direct system X i i∈I of τ-torsion-free modules (3) Tor R n (A, X + ) � Ext n R (A, X + ) for any n-presented right R-module A and each τ-torsion-free module X (4) Tor R n (A, E + ) � 0 for any n-presented right R-module
. en, K n−1 is finitely generated, and we get an exact sequence of right R-modules 0 ⟶ K n−1 ⟶ F n−1 ⟶ K n−2 ⟶ 0. Let E i i∈I be any direct system of τ-torsion-free injective right R-modules (with I directed). en, lim 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 X i i∈I be any direct system of τ-torsion-free modules (with I directed). en, we have a commutative diagram with exact rows: where E(X i ) is the injective hull of X i . Since K n−1 is finitely generated, by 24.9 in [10], the maps ϕ 1 , ϕ 2 , and ϕ 3 are monic. Since τ is a hereditary torsion theory and X i is τ-torsion-free, by Proposition 3.2, Chap VI in [9], E(X i ) is τ-torsion-free. And so, by the above proof, ϕ 2 is an isomorphism. Hence, ϕ 1 is also an isomorphism by the Five Lemma again, and then K n−1 is τ-finitely presented by Proposition 2.5 (3) [4], and thus A is τ-(n + 1)-presented. erefore, 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 X i i∈I be a direct system of τ-torsionfree n-FP-injective modules. en by Proposition 1 in [7], we have a map-pure, and hence pure exact sequence 0 ⟶ K ⟶ ⊕ i∈I X i ⟶ lim ⟶ X i ⟶ 0. Observing that ⊕ i∈I X i is τ-torsion-free and n-FP-injective, by (10), we have that lim Journal of Mathematics We call a right R-module X weakly n-FP-injective if Ext n R (A, X) � 0 for any (n + 1)-presented right R-module A. Let τ � (0, mod − R). en, we have the following results.

Corollary 2.
Let n be a positive integer. en, the following statements are equivalent for a ring R: for any n-presented right R-module A and direct system X i i∈I of right R-modules (3) Tor R n (A, X + ) � Ext n R (A, X) + for any n-presented right R-module A and each right R-module X (4) Tor R n (A, E + ) � 0 for any n-presented right R-module A and each injective right R-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-FPinjective (7

) Any direct limit of FP-injective modules is n-FPinjective (8) Any direct limit of injective modules is n-FP-injective (9) A right R-module X is n-FP-injective if and only if X + is n-flat (10) 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 eorem 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 eorem 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-FPinjective. □ Let n � 1; then, by eorem 2, we can obtained several results on right τ-coherent rings.

Corollary 4. Let τ be a hereditary torsion theory of mod-R.
en, the following statements are equivalent for the ring R: (1) R is right τ-coherent (2) lim ⟶ Ext 1 R (A, X i ) � Ext 1 R (A, lim ⟶ X i ) for any finitely presented right R-module A and direct system X i i∈I of τ-torsion-free modules (3) Tor R 1 (A, X + ) � Ext n R (A, X) + for any finitely presented right R-module A and each τ-torsion-free module X (4) Tor R 1 (A, E + ) � 0 for any finitely presented right R-module A and each τ-torsion-free injective module E (5) If X is a τ-1-FP-injective module, then X is FPinjective (6) Any direct limit of τ-torsion-free FP-injective modules is FP-injective (7) Any direct limit of τ-torsion-free injective modules is FP-injective. (

8) A τ-torsion-free module X is FP-injective if and only if
X + is flat (9) 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
e authors declare that they have no conflicts of interest.