STRONGLY J-N-COHERENT RINGS

. Let R be a ring and n a ﬁxed positive integer. A right R -module M is called strongly J - n -injective if every R -homomorphism from an n -generated small submodule of a free right R -module F to M extends to a homomorphism of F to M ; a right R -module V is said to be strongly J - n -ﬂat, if for every n - generated small submodule T of a free left R -module F , the canonical map V ⊗ T → V ⊗ F is monic; a ring R is called left strongly J - n -coherent if every n -generated small submodule of a free left R -module is ﬁnitely presented; a ring R is said to be left J - n -semihereditary if every n -generated small left ideal of R is projective. We study strongly J - n -injective modules, strongly J - n -ﬂat modules and left strongly J - n -coherent rings. Using the concepts of strongly J - n -injectivity and strongly J - n -ﬂatness of modules, we also present some characterizations of strongly J - n -coherent rings and J - n -semihereditary rings.


Introduction
Throughout this paper, m and n are positive integers unless otherwise specified, R is an associative ring with identity, I is an ideal of R, J = J(R) is the Jacobson radical, and all modules considered are unitary.
Left coherent rings, left semihereditary rings and their generalizations have been studied by many authors.For example, a ring R is said to be left J-coherent [6] (resp., left J-semihereditary [6]) if every finitely generated left ideal in J(R) is finitely presented (resp., projective); a ring R is said to be left n-coherent [13] (resp., left n-semihereditary [18,19]) if every n-generated left ideal of R is finitely presented (resp., projective).By [19,Theorem 1], a ring R is left n-semihereditary if and only Lemma 2.1.Let U R ≤ U R .Then the following statements are equivalent: (1) U is J-(n, ∞)-pure in U.
(2) For every finitely generated free right R-module F and each n-generated small submodule T of F, the canonical map Hom R (F/T, U ) → Hom R (F/T, U/U ) is surjective.Lemma 2.2.Let U R ≤ U R .Then the following statements are equivalent: (1) U is J-(∞, n)-pure in U.
(2) For every finitely generated small submodule T of R n R , the canonical map Hom R (R n /T, U ) → Hom R (R n /T, U/U ) is surjective.
Recall that a right R-module M is called I-(m, n)-injective [21], if every Rhomomorphism from an n-generated submodule T of I m to M extends to one from R m to M .A right R-module M is called I-n-injective [20] if it is I-(1, n)injective.Inspired by these concepts, we introduce the concept of strongly J-ninjective modules as follows.
Definition 2.3.A right R-module M is called strongly J-n-injective if every Rhomomorphism from an n-generated small submodule of a free right R-module F to M extends to a homomorphism of F to M .A right R-module M is called J-FPinjective if every R-homomorphism from a finitely generated small submodule of a free right R-module F to M extends to a homomorphism of It is easy to see that a right R-module M is strongly J-n-injective if and only if it is J-(m, n)-injective for every positive integer m; a right R-module M is J-FPinjective if and only if it is strongly J-n-injective for every positive integer n.Theorem 2.4.Let M be a right R-module.Then the following statements are equivalent: (1) M is strongly J-n-injective.
(2) Ext 1 (F/T, M ) = 0 for every free right R-module F and every n-generated small submodule T of F.
(3) Ext 1 (F/T, M ) = 0 for every finitely generated free right R-module F and every n-generated small submodule T of F.

Proof.
Let N be a J-(n, ∞)-pure submodule of a strongly J-n-injective right Rmodule M .For any n-generated small submodule T of a finitely generated free right R-module F , we have the exact sequence and so N is strongly J-n-injective.
Theorem 2.6.The following statements are equivalent for a ring R: (1) R is right strongly J-n-injective.
(2) Every finitely generated small submodule T of the left R-module R n is a left annihilator of a subset X of R n .
(3) If r Rn (T ) ⊆ r Rn (α) for a finitely generated small submodule T of the left R-module R n and α ∈ R n , then α ∈ T .
(4) R n /T is a torsionless left R-module for every finitely generated small submodule T of R n .
(5) l R n r Rn (T ) = T for every finitely generated small submodule T of the left R-module R n .
(2) ⇔ (4) ⇔ (5) follows from [22,Lemma 2.3]. (5 be an n-generated small submodule of F , and f be a right R-homomorphism from K to R. Write Then T is a small submodule of the left R-module R n and r Rn (T ) ⊆ r Rn (α).
and therefore g extends f .
Recall that a ring R is called semiregular [11] if for any a ∈ R, there exists e 2 = e ∈ aR such that (1 − e)a ∈ J(R).By [11,Theorem B.44 and by [11,Theorem B.54], direct sums and direct summands of semiregular modules are semiregular.We recall also that a right R-module M is called strongly n-injective [22] if every R-homomorphism from an n-generated submodule of a free right R-module F to M extends to a homomorphism of F to M .Proposition 2.7.If R is a semiregular ring, then a right R-module M is strongly n-injective if and only if it is strongly J-n-injective.
Proof.Necessity is clear.To prove the sufficiency, let N be an n-generated submodule of a finitely generated free right R-module F and f : N → M be a right R-homomorphism.Since R is semiregular, by [11,Lemma B.54], F is semiregular.So, by [11,Lemma B.51], F = P ⊕ K, where P is projective, P ⊆ N and Then h is a well-defined left R-homomorphism and h extends f .

Strongly J-n-flat modules
Recall that a right R-module V is said to be n-flat [13,7], if for every n-generated V is said to be J-flat [6], if for every finitely generated left ideal T in J(R), the is monic.Inspired by these concepts, we introduce the concepts of strongly J-n-flat modules and strongly J-flat modules as follows.
Definition 3.1.A right R-module V is said to be strongly J-n-flat, if for every ngenerated small submodule T of a free left R-module F , the canonical map V ⊗T → V ⊗ F is monic.A right R-module V is said to be strongly J-flat if it is strongly J-n-flat for every positive integer n.Theorem 3.2.For a right R-module V , the following statements are equivalent: (1) V is strongly J-n-flat.
(2) Tor 1 (V, F/L) = 0 for every finitely generated free left R-module F and any n-generated small submodule L of F.
(3) Tor 1 (V, F/L) = 0 for every free left R-module F and any n-generated small submodule L of F.
(6) For every finitely generated small submodule T of the right R-module R n and any homomorphism f : R n /T → V , f factors through a finitely generated free right R-module F, that is, there exist a homomorphism g : R n /T → F and a homomorphism h : F → V such that f = hg.
(2) ⇔ (4) follows from the isomorphism Tor By (2), the canonical map U ⊗ F/L → U ⊗ F/L is monic for any finitely generated free left R-module F and any n-generated small submodule L of F , and so U is where F 1 is free.Then by (5), that Tor R 1 (V, F/L) = 0 for every finitely generated free left R-module F and any n-generated small submodule L of F .
(5) ⇒ (6) Let 0 → K → F 1 → V → 0 be an exact sequence of right R-modules, where F 1 is free.Then by (5), K is J-(∞, n)-pure in F 1 .And so, by Lemma 2.2, we have that the canonical map Hom(R n /T, F 1 ) → Hom(R n /T, V ) is surjective for any finitely generated small submodule T of R n R .This follows that f factors through a finitely generated free right R-module F since R n /T is finitely generated.
with U J-n-flat.Then for any finitely generated small submodule T of R n R and any homomorphism f : R n /T → V , by (6), there exist a finitely generated free module F , two homomorphisms g ∈ Hom R (R n /T, F ) and h ∈ Hom R (F, V ) such that f = hg.Since F is projective, there exists a homomorphism α : (7) ⇒ (8) Let g : M → V be an epimorphism and f : R n /T → V be any homomorphism, where T is a finitely generated small submodule of R n .By (7), f factors through a finitely generated projective right R-module P , i.e., there exist ϕ : R n /T → P and ψ : F → V such that f = ψϕ.Since P is projective, there exists a homomorphism θ : P → M such that ψ = gθ.Now write h = θϕ, then h is a homomorphism from R n /T to M , and f = ψϕ = g(θϕ) = gh.And so (8) follows.
(8) ⇒ (7) Let F 1 be a free module and π : F 1 → V be an epimorphism.By (8), there exists a homomorphism g : R n /T → F 1 such that f = πg.Note that Im(g) is finitely generated, so there is a finitely generated free module F such that Im(g) ⊆ F ⊆ F 1 .Let ι : F → F 1 be the inclusion map and h = πι.Then h is a homomorphism from F to V and f = hg.Corollary 3.3.For a right R-module V , the following statements are equivalent: (1) V is strongly J-flat.
(2) Tor 1 (V, F/L) = 0 for every finitely generated free left R-module F and any finitely generated small submodule L of F.
(3) Tor 1 (V, F/L) = 0 for every free left R-module F and any finitely generated small submodule L of F.
(6) For every finitely generated small submodule T of a finitely generated free right R-module F , any homomorphism f : F/T → V factors through a finitely generated free right R-module F 1 , that is, there exist a homomorphism g : F/T → F 1 and a homomorphism h : F 1 → V such that f = hg.
Then by the strongly J-n-flatness of V , there exist positive integer l, U ∈ V l and C ∈ R l×n such that CA = 0 and X = U C. Since K is J-(n, ∞)-pure in V and hence J-(n, l)-pure, by [21, Theorem 2.4(3)], we have Remark 3.6.From Theorem 3.2, the strongly J-n-flatness of V R can be characterized by the strongly J-n-injectivity of V + .On the other hand, by [5, Lemma 2.7(1)], the sequence Tor 1 (V + , M ) → Ext 1 (M, V ) + → 0 is exact for all finitely presented left R-module M , so if V + is strongly J-n-flat, then V is strongly J-n-injective.

Proposition 3.7. If R is a semiregular ring, then a left R-module M is strongly n-flat if and only if it is strongly J-n-flat.
Proof.Theorem 3.2(4), Proposition 2.7 and [22, Theorem 3.1(4)] give the desired result.

Strongly J-n-coherent rings
Recall that a ring R is called left (m, n)-coherent [17] if every n-generated submodule of R m is finitely presented; a ring R is called left J-coherent [6] if every finitely generated left ideal in J(R) is finitely presented; a ring R is called left J-ncoherent [20] if every n-generated left ideal in J(R) is finitely presented.Inspired by these concepts, we introduce the concepts of strongly J-n-coherent rings and J-(m, n)-coherent rings as follows.
Definition 4.1.A ring R is called left strongly J-n-coherent if every n-generated small submodule of a free left R-module is finitely presented.A ring R is called left J-(m, n)-coherent if every n-generated small submodule of R R m is finitely presented.
Recall that a left R-module A is called 2-presented if there exists an exact sequence F 2 → F 1 → F 0 → A → 0 in which every F i is a finitely generated free module.It is easy to see that a ring R is left strongly J-n-coherent if and only if it is left J-(m, n)-coherent for all positive integers m, if and only if every J-(m, n)presented left R-module is 2-presented for all positive integers m.Theorem 4.2.Let R be a ring.Then the following statements are equivalent: (1) R is a left J-coherent ring.
(2) Every finitely generated small submodule A of a finitely generated free left R-module F is finitely presented.
(3) Every finitely generated small submodule A of a free left R-module F is finitely presented.
(4) Every finitely generated small submodule A of a projective left R-module F is finitely presented.
(5) For every finitely generated free left R-module F and any finitely generated A is a finitely generated left ideal in J(R), by hypothesis, A is finitely presented.
Assume that every finitely generated small submodule of the left R-module R m−1 is finitely presented.Then for any finitely generated small submodule A of the left , where e j ∈ R m with 1 in the jth position and 0's in all other positions.Then each a ∈ A has a unique expression a finitely generated left ideal in J(R).By hypothesis, L is finitely presented, and so B is finitely generated.Since B is isomorphic to a small submodule of R m−1 , the induction hypothesis gives B is finitely presented.Therefore, A is also finitely presented by [16, 25.  (1) R is left strongly J-n-coherent. ( → T is an exact sequence of left R-modules, where T is a finitely generated small submodule of a free left R-module, then K is finitely generated. (3) l R n (X) is a finitely generated submodule of R R n for any finite subset X of (4) For any finitely generated small submodule S of the right R-module R n , the dual module (R n /S) * is a finitely generated left R-module. Proof.
(1) ⇒ (2) Since R is left strongly J-n-coherent and im(g) is an n-generated small submodule of a free left R-module, im(g) is finitely presented.Noting that the sequence 0 → ker(g) → R n → im(g) → 0 is exact, we have that ker(g) is finitely generated.Thus K im(f ) = ker(g) is finitely generated. ( Then we have an exact sequence of left R- is a finitely generated left R-module. ( Then we have an exact sequence of left R-modules Let F be a class of right R-modules and M a right R-module.Following [9], we say that a homomorphism ϕ : An F-preenvelope ϕ : M → F is said to be an F-envelope if every endomorphism g : F → F such that gϕ = ϕ is an isomorphism.Dually, we have the definitions of an F-precover and an F-cover.F-envelopes (F-covers) may not exist in general, but if they exist, they are unique up to isomorphism.
Theorem 4.5.The following statements are equivalent for a ring R: (1) R is left strongly J-n-coherent.
(2) lim −→ Ext 1 (F/T, M α ) Ext 1 (F/T, lim −→ M α ) for every n-generated small submodule T of a finitely generated free left R-module F and direct system (M α ) α∈A of left R-modules.
(3) Tor 1 ( N α , F/T ) Tor 1 (N α , F/T ) for any family {N α } of right Rmodules and any n-generated small submodule T of a finitely generated free left R-module F.
(4) Any direct product of copies of R R is strongly J-n-flat.
(5) Any direct product of strongly J-n-flat right R-modules is strongly J-n-flat.
(6) Any direct limit of strongly J-n-injective left R-modules is strongly J-ninjective.
(7) Any direct limit of injective left R-modules is strongly J-n-injective.
(8) A left R-module M is strongly J-n-injective if and only if M + is strongly J-n-flat.
(7) ⇒ (1) Let F be a finitely generated free left R-module and T be an ngenerated small submodule of F , and let (E α ) α∈A be a direct system of injective left R-modules (with A directed).Then lim −→ E α is strongly J-n-injective by (7), and so Ext 1 (F/T, lim −→ M α ) = 0. Thus we have a commutative diagram with exact rows: Since f and g are isomorphism by [16, 25.4(d)], h is an isomorphisms by the Five Lemma.Now, let (M α ) α∈A be any direct system of left R-modules (with A directed).Then we have a commutative diagram with exact rows: where E(M α ) is the injective hull of M α .Since T is finitely generated, by [16, 24.9], the maps φ 1 , φ 2 and φ 3 are monic.By the above proof, φ 2 is an isomorphism.Hence φ 1 is also an isomorphism by Five Lemma again, so T is finitely presented by [16, 25.4(d)] again.Therefore R is left strongly J-n-coherent.
(4) ⇒ (1) Let T be an n-generated small submodule of a finitely generated free left R-module F .By (4), Tor 1 (ΠR, F/T ) = 0. Thus we have a commutative diagram with exact rows:   (4) Any direct product of copies of R R is strongly J-flat.
(5) Any direct product of strongly J-flat right R-modules is strongly J-flat.
(6) Any direct limit of J-FP-injective left R-modules is J-FP-injective.
(7) Any direct limit of injective left R-modules is J-FP-injective.
(8) A left R-module M is J-FP-injective if and only if M + is strongly J-flat.
(9) A left R-module M is J-FP-injective if and only if M ++ is J-FP-injective.
(10) A right R-module M is strongly J-flat if and only if M ++ is strongly J-flat.Proof.We need only to proof the sufficiency.Let R be left strongly J-n-coherent.
Then by Theorem 4.5(4), any direct product of copies of R R is strongly J-n-flat.
Note that R is a semiregular ring, by Proposition 3.7, any direct product of copies of R R is strongly n-flat.And so, by [22, Theorem 4.2(4)], R is left strongly ncoherent.
Corollary 4.8.Let R be a left strongly J-n-coherent ring.Then every left Rmodule has a strongly J-n-injective cover.
Proof.Let 0 → A → B → C → 0 be a pure exact sequence of left R-modules with B strongly J-n-injective.Then 0 → C + → B + → A + → 0 is split.Since R is left strongly J-n-coherent, B + is strongly J-n-flat by Theorem 4.5 (8), so C + is strongly J-n-flat, and hence C is strongly J-n-injective by Remark 3.4.Thus, the class of strongly J-n-injective modules is closed under pure quotients, and so by [10, Theorem 2.5], every left R-module has a strongly J-n-injective cover.
Proposition 4.9.Let R be a left J-coherent ring.Then every left R-module has a J-FP-injective cover.
Proof.It is similar to the proof of Corollary 4.8.

( 7 )( 8 )
For every finitely generated small submodule T of the right R-module R n and any homomorphism f : R n /T → V , f factors through a finitely generated projective right R-module P. For every finitely generated small submodule T of the right R-module R n , if g : M → V is an epimorphism, then for any homomorphism f : R n /T → V , there exists a homomorphism h : R n /T → M such that f = gh.Proof.(1) ⇔ (2) follows from the exact sequence 0 → Tor 1

( 7 )( 8 )
For every finitely generated small submodule T of a free right R-module F , any homomorphism f : F/T → V factors through a finitely generated projective right R-module P. For every finitely generated small submodule T of a free right R-module F , if g : M → V is an epimorphism, then for any homomorphism f : F/T → V , there exists a homomorphism h : F/T → M such that f = gh.Proposition 3.4.Every J-(n, ∞)-pure submodule of a strongly J-n-flat module is strongly J-n-flat.

Corollary 3 . 5 .
flat for any positive integer m by [21, Theorem 4.2(5)], and hence K is strongly J-n-flat.Every J-pure submodule of a strongly J-flat module is strongly J-flat.

Remark 4 . 3 .Theorem 4 . 4 .
By Theorem 4.2, it is easy to see that R is left J-coherent if and only if R is left strongly J-n-coherent for each positive integer n.The following statements are equivalent for a ring R:

( 10 )( 11 )( 12 )( 13 )
A right R-module M is strongly J-n-flat if and only if M ++ is strongly J-n-flat.For any ring S, Tor 1 (Hom S (B, C), F/T ) Hom S (Ext 1 (F/T, B), C) for the situation ( R (F/T ), R B S , C S ) with F a finitely generated free left R-module and T an n-generated small submodule of F and C S injective.Every right R-module has a strongly J-n-flat preenvelope.For every finitely generated small submodule S of the right R-module R n , the right R-module R n /S has a finitely generated projective preenvelope.Proof.(1) ⇒ (2) follows from [5, Lemma 2.9(2)].(1)⇒ (3) follows from[5, Lemma 2.10(2)].

Since f 2
and f 3 are isomorphisms by[16, 25.4(g)], f 1 is an isomorphism by the Five Lemma.So T is finitely presented by[16, 25.4(g)] again.Hence R is left strongly J-n-coherent.

( 5 )
⇒ (12)  Let N be any right R-module.By[9, Lemma 5.3.12],there is a cardinal number ℵ α dependent on Card(N ) and Card(R) such that for any homomorphism f : N → F with F strongly J-n-flat, there is a pure submodule S of F such that f (N ) ⊆ S and Card S ≤ ℵ α .Thus f has a factorization N → S → F with S strongly J-n-flat by Proposition 3.3.Now let {ϕ β } β∈B be all such homomorphisms ϕ β : N → S β with Card S β ≤ ℵ α and S β strongly J-n-flat.Then any homomorphism N → F with F strongly J-n-flat has a factorization N → S i → F for some i ∈ B. Thus the homomorphism N → Π β∈B S β induced by all ϕ β is a strongly J-n-flat preenvelope since Π β∈B S β is strongly J-n-flat by(5).

( 11 )( 12 )( 13 )Corollary 4 . 7 .
For any ring S, Tor 1 (Hom S (B, C), F/T ) Hom S (Ext 1 (F/T, B), C) for the situation ( R (F/T ), R B S , C S ) with F a finitely generated free left R-module and T a finitely generated small submodule of F and C S injective.Every right R-module has a strongly J-flat preenvelope.For every finitely generated small submodule S of a finitely generated right R-module F , the right R-module F/S has a finitely generated projective preenvelope.Let R be a semiregular ring.Then it is left strongly n-coherent if and only if it is left strongly J-n-coherent.
[11,le M is called semiregular if for any m ∈ M , we have M = P ⊕ K, where P is projective, P ⊆ mR, and mR ∩ K is small in K.It is easy to see that a ring R is semiregular if and only if the right R-module R R is semiregular.By[11,   Theorem B.51], a module M is semiregular if and only if, for any finitely generated submodule N of M , we have M = P ⊕ K, where P is projective, P ⊆ N , and