( n, d ) -COCOHERENT RINGS, ( n, d ) -COSEMIHEREDITARY RINGS AND ( n, d ) - V -RINGS

. Let R be a ring, n be an non-negative integer and d be a positive integer or ∞ . A right R -module M is called ( n,d ) ∗ -projective if Ext 1 R ( M,C ) = 0 for every n -copresented right R -module C of injective dimension ≤ d ; a ring R is called right ( n,d ) -cocoherent if every n -copresented right R -module C with id ( C ) ≤ d is ( n +1)-copresented; a ring R is called right ( n,d ) -cosemihereditary if whenever 0 → C → E → A → 0 is exact, where C is n -copresented with id ( C ) ≤ d , E is ﬁnitely cogenerated injective, then A is injective; a ring R is called right ( n,d ) - V -ring if every n -copresented right R -module C with id ( C ) ≤ d is injective. Some characterizations of ( n,d ) ∗ -projective modules are given, right ( n,d )-cocoherent rings, right ( n,d )-cosemihereditary rings and right ( n,d )- V -rings are characterized by ( n,d ) ∗ -projective right R -modules. ( n,d ) ∗ -projective dimensions of modules over right ( n,d )-cocoherent rings are investigated.


Introduction
Throughout this paper, R is an associative ring with identity and all modules considered are unitary, n is a non-negative integer, d is a positive integer or ∞ unless a special note.
In 1982, V. A. Hiremath [4] defined and studied finitely corelated modules. Following [4], a right R-module M is said to be finitely corelated if there is a short exact sequence 0 → M → N → K → 0 of right R-modules with N finitely cogenerated, cofree and K is finitely cogenerated, where a right R-module N is said to be cofree if it is isomorphic to a direct product of the injective hulls of some simple right R-modules. Finitely corelated modules are also called finitely copresented modules  in some literatures such as [7]. Following [12], a right R-module M is said to be FCP-projective if Ext 1 R (M, C) = 0 for every finitely copresented right R-module C. In [12], right V -rings are characterized by FCP-projective right R-modules. We recall also that R is called right co-semihereditary [6,8,12] if every finitely cogenerated factor module of a finitely cogenerated injective right R-module is injective, R is called right co-coherent [12] if every finitely cogenerated factor module of a finitely cogenerated injective right R-module is finitely copresented. It is easy to see that right V -rings, right co-semihereditary rings and right co-coherent rings are the dual concepts of von Neumann regular rings, right semihereditary rings and right coherent rings. In this paper, right cocoherent rings will denote right co-coherent rings in order to facilitate. In [12], right V -rings, right co- In 1999, Xue introduced n-copresented modules and n-cocoherent rings respectively in [9]. According to [9], M is said to be n-copresented if there is an exact is a finitely cogenerated injective module. It is easy to see that a module M is finitely cogenerated if and only if it is 0-copresented, a module M is finitely copresented if and only if it is 1-copresented. We call any module (−1)-copresented.
n-copresented modules have been studied in [2,9,11]; R is called right n-cocoherent [9] in case every n-copresented right R-module is (n + 1)-copresented. It is easy to see that R is right cocoherent if and only if it is right 1-cocoherent. Following [5], a ring R is called right co-noetherian if every factor module of a finitely cogenerated right R-module is finitely cogenerated. By [4,Proposition 17], a ring R is right co-noetherian if and only if it is right 0-cocoherent. In [11], we extend the concepts of FCP-projective modules, cosemihereditary rings and V -rings to (n, d)projective modules, n-cosemihereditary rings and n-V -rings respectively, right n-V -rings and right n-cosemihereditary rings are characterized by (n, 0)-projective right R-modules, (n, 0)-projective dimensions of right R-modules over right ncocoherent rings are investigated. Following [11], a right R-module M is called (n, d)-projective if Ext d+1 R (M, A) = 0 for every n-copresented right R-module A; a ring R is called right n-cosemihereditary if every submodule of a projective right and n-V -rings can be found in [11,Theorem 3.7] and [11,Theorem 3.9], respectively.

(n, d) * -Projective modules and (n, d)-cocoherent rings
We start with the following definition.  (1) It is easy to see that if a module M is (n, d) * -projective, then it is (n , d ) * -projective for any n ≥ n and d ≤ d.
Recall that a short exact sequence of right R-modules 0 → A → B → C → 0 is said to be n-copure [11] if every n-copresented module is injective with respect to this sequence.
is said to be (n, d)-copure if every n-copresented module with injective dimension ≤ d is injective with respect to this sequence.
Theorem 2.5. Let M be a right R-module. Then the following statements are equivalent: (3) If E is an (n − 1)-copresented factor module of a finitely cogenerated injec- Proof.
0 of right R-modules with P (n, d) * -projective, and so, for each n-copresented This implies that Ext 1 R (M, C) = 0, and therefore (1) follows.
(1) ⇒ (7) ⇒ (8) ⇒ (1) are similar to the proofs of (1) (1) It is easy to see that if a ring R is right (n, d)-cocoherent, then it is right (n , d )-cocoherent for any n ≥ n and d ≤ d.
(3) A ring R is right n-cocoherent if and only if it is right (n, ∞)-cocoherent.
Lemma 2.9. Let R be a right (n,d)-cocoherent ring and M a right R-module. Then the following statements are equivalent: where E is finitely cogenerated injective, and E is (n−1)-copresented. Since copresented, so E is n-copresented, and thus Ext k+1 R (M, A) ∼ = Ext k R (M, E ) = 0 by induction hypothesis.
for any ncopresented modules C with id(C) ≤ d.
Proof. Since R is right (n, d)-cocoherent and P 0 , P 1 , . . . , P k−1 are (n, d)-projective, by Corollary 2.10, we have Theorem 2.12. Let R be a right (n, d)-cocoherent ring and M be a right R-module.
Then the following statements are equivalent: (2) Ext k+l R (M, C) = 0 for all n-copresented modules C with id(C) ≤ d and all positive integers l.

(n, d)-Cosemihereditary rings and (n, d)-V -rings
As the beginning of this section, we extend the concept of n-cosemihereditary rings as follows.  (1) It is easy to see that if a ring R is right (n, d)-cosemihereditary, then it is right (n , d )-cosemihereditary for any n ≥ n and d ≤ d.
(3) Ext 2 R (M, C) = 0 for any right R-module M and any n-copresented right R-module C with id(C) ≤ d.
Proof. (1) ⇒ (2) Let C be an n-copresented right R-module with injective dimension ≤ d. Then there exists an exact sequence 0 → C → E → E → 0, where E is finitely cogenerated injective, E is (n − 1)-copresented and id(E ) ≤ d − 1. Since R is right (n, d)-cosemihereditary, E is finitely cogenerated injective, and so C is (n + 1)-copresented, it shows that R is right (n, d)-cocoherent. Now let M be a right R-module. Then for any n-copresented right R-module C with id(C) ≤ d,   to E . Then for any projective right R-module P and any submodule K of P , K is (n, d) * -projective by (4). So for any n-copresented right R-module C with id(C) ≤ d, we have an exact sequence 0 = Ext 1 R (K, C) → Ext 2 R (P/K, C) → Ext 2 R (P, C) = 0, which implies that Ext 2 R (P/K, C) = 0. Note that Ker(f ) is n-copresented and id(Ker(f )) ≤ d, we get an exact sequence 0 = Ext 1 R (P/K, E) → Ext 1 R (P/K, E ) → Ext 2 R (P/K, Ker(f )) = 0, and then Ext 1 R (P/K, E ) = 0, which shows that E R is P Rinjective from the exact sequence Hom(P, E ) → Hom(K, E ) → Ext 1 R (P/K, E ). Therefore, E is injective.  (1) R is a right n-cosemihereditary ring.
(3) Ext 2 R (M, C) = 0 for any right R-module M and any n-copresented right R-module C.
Corollary 3.5. The following statements are equivalent for a ring R: (1) R is a right cosemihereditary ring.
(3) Ext 2 R (M, C) = 0 for any right R-module M and any finitely copresented right R-module C.
(4) Every submodule of an FCP-projective right R-module is FCP-projective.
(5) Every submodule of a projective right R-module is FCP-projective.
Corollary 3.6. The following statements are equivalent for a ring R: (1) R is a right cohereditary ring.
(3) Ext 2 R (M, C) = 0 for any right R-module M and any finitely cogenerated right R-module C.
(5) Every submodule of a projective right R-module is FCG-projective.
Next we extend the concept of right n-V -rings as follows.  (1) It is easy to see that if n ≥ n and d ≤ d, then a right (n, d)-V -ring is a right (n , d )-V -ring.
(2) A ring R is a right n-V -ring if and only if it is a right (n, ∞)-V -ring. Now we give some characterizations of right (n, d)-V -rings.
Theorem 3.9. The following conditions are equivalent for a ring R: (1) R is a right (n, d)-V -ring.
(5) ⇒ (6) Let C be an n-copresented right R-module with id(C) ≤ d. Then there exists an exact sequence 0 → C → E → E → 0 of right R-modules, where E is finitely cogenerated injective, E is (n − 1)-copresented and id(E ) ≤ d − 1. By (5), E is (n, d) * -projective, so E is projective respect to this exact sequence by Theorem 2.5(3). This follows that C is isomorphic to a direct summand of E, and therefore C is injective.
Recall that a right R-module M is called FCG-projective [11]  (1) R is a right V -ring.
(4) Every right R-module is FCG-projective.      (11) R is right cocoherent and for every finitely cogenerated injective right Rmodule E, every finitely copresented factor module E of E is FCP-projective.  (1) ⇒ (8) Let R be a right V -ring. Then every simple right R-module is injective.
For any finitely cogenerated right R-module M , we have E(M ) ∼ = E(S 1 ) + · · · + E(S n ) for some finite set S 1 , . . . , S n of simple modules by [1,Proposition 18.18], so E(M ) ∼ = S 1 + · · · + S n is semisimple. Thus M is a direct summand of E(M ), and therefore M is injective.
(13) ⇒ (1) Let S be any simple right R-module. Suppose S is not injective. Let x ∈ E(S)\S and let A be a submodule of E(S) maximal with respect to S ⊆ A and x / ∈ A, then 0 = x + A ∈ ∩{K ≤ E(S)/A | K = 0}, which implies that E(S)/A is finitely cogenerated and whence A is finitely copresented. By (13), A is injective. It follows that A = E(S), which contradicts the fact that x / ∈ A. Hence S is injective and so R is a right V -ring. Recall that a right R-module M is called n-presented [3] if there is an exact sequence of right R-modules F n → F n−1 → · · · → F 1 → F 0 → M → 0 where each F i is a finitely generated free, equivalently projective right R-module; a left R-module M is called (n, 0)-flat [10] if Tor R 1 (A, M ) = 0 for every n-presented right R-module A. A ring R is called right n-regular [10] if every n-presented right Rmodule is projective. By [10, Theorem 3.9], a ring R is right n-regular if and only if every left R-module M is (n, 0)-flat.
Proof. Let M be an (n, 0)-projective module. To prove M is (n, 0)-flat, we need prove Tor R 1 (A, M ) = 0 for every n-presented R-module A. Since A is n-presented, Hom R (A, E(S)) is n-copresented for any simple module S. Let 0 → K → P → M → 0 be an exact sequence of R-modules with P projective. Then by Theorem 2.5, this exact sequence is n-copure. And so we get an exact sequence of R-modules 0 → Hom R (M, Hom R (A, E(S))) → Hom R (P, Hom R (A, E(S))) → Hom R (K, Hom R (A, E(S))) → 0.
Let S 0 denote an irredundant set of representatives of the simple R-modules and let C = S∈S0 E(S). Then by [1,Corollary 18.16], C is a cogenerator. And we have an exact sequence of R-modules 0 → Hom R (M ⊗ R A, C) → Hom R (P ⊗ R A, C) → Hom R (K ⊗ R A, C) → 0.
So, by [1,Proposition 18.14], the sequence of R-modules is exact. This shows that Tor R 1 (A, M ) = 0, as required.
Corollary 3.12. Let R be a commutative n-V -ring. Then it is an n-regular ring.
The following result is well-known.
Corollary 3.13. Let R be a commutative V -ring. Then it is a regular ring.