ON FI-COTORSION MODULES AND DIMENSIONS

unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract. In this paper we study the class of FI-cotorsion modules and we introduce there dimensions of modules and rings. An R-module M is called FI-cotorsion if Ext1 R(F,M)= 0 for any FI-flat R-module F . Also, we investigate some properties of FI-cotorsion modules and FI-cotorsion envelopes and we give a characterization of IF-ring. Then, we study the FI-flat and the FI-cotorsion envelope and we show when they existed. Furthermore, we present the notation of FI-cotorsion dimension of modules and rings.


INTRODUCTION
We first recall some known notions and facts that well needed later. Now, we recall the notion of (pre-)cover and (pre-)envelope of modules. Definition 1. [4,11] Let M be any class of modules.

Definition 2. [4]
Let M and N be two classes of modules. (2) M is called a special preenvelope (resp., precover) class if every R-module has a special preenvelope (resp., precover).
(3) M is called a cover (resp., envelope) class if every R-module has a M -cover (resp., M -envelope).
(4) A cotorsion theory (M ,N ) is called complete if every module has a special M −precover and has a special N −preenvelope. The concept of cotorsion modules is introduced by Enochs (1984) [3] defined by: After in (2005) Ding and Mao [9] introduced the cotorsion dimensions of modules and rings.
On the other hand, Mao and Ding (2007) [8] introduced and studied the concept of FI-flat module defined by: Later, Selvaraj et al. (2017) [2] give the definition of FI-cotorsion modules (Definition5).
Motivated by [1,9], we study the FI-cotorsion modules and we introduce the FI-cotorsion dimensions of modules and of rings. This paper is organized as follows: In Section 2, we give a characterization of FI-cotorsion modules and present several properties. We show that this class behave in short exact sequence and stable under direct product.
Also, we give sufficient conditions such that every cotorsion module is FI-cotorsion and such that every FI-cotorsion module is injective. Finally, we prove, for a module M, when the character module M + is FI-cotorsion.
In Section 3, We show that (F F , F C ) is a cotorsion theory and also hereditary cotorsion theory, where F F (resp., F C ) denote the class of FI-flat (resp., FI-cotorsion) modules. Also, we discus when a module has a FI-flat cover and FI-cotorsion envelope and we prove that the FI-flat envelope of a module M is FI-cotorsion if and only if M is also FI-cotorsion.
In Section 4, we introduce the concept FI-cotorsion dimension of modules and rings. We give characterizations of these dimensions and we study the FI-cotorsion dimension of modules under short exact sequences and direct product of modules. Finally, We prove that, the FIcotorsion of a ring R is ≤ 1 if and only if every FI-flat R-module has a projective dimension at most 1.

FI-COTORSION MODULES
In this section we study the class of FI-cotorsion modules and we give a large number of proberties.
for any FI-flat R-module F. The class of all FI-cotorsion module is denoted by F C .
Remark 1. From the definition above, it's easy to see that we have the following inclusions between classes of modules: The following proposition gives a characterization of FI-cotorsion modules. Proposition 1. For an R-module M, the following statements are equivalent: (1) M is FI-cotorsion.
(2) Ext n R (F, M) = 0 for every n ≥ 1 and for every FI-flat R-module F.    Proof. 1) ⇒ 2) We prove it by induction on n. If n = 1, then its true by definition. Suppose that it true for n − 1 and we prove it for n. Let F be FI-flat module and let the exact sequence of R -modules 0 → K → F 0 → F → 0, where F 0 is free and it's easy to see that K is also FI-flat.
Applying the long exact sequence of the functor Hom(., M) we get the exact sequence: where both ends vanish since F 0 is free. Now since K is FI-flat and by induction we have 3) ⇔ 1) Let F be a FI-flat R-module. Consider the short exact sequence of R -modules 1) ⇒ 5) Suppose that M is a FI-cotorsion R-module and consider the short exact of R-modules Applying the long exact sequence of the functor  (1) M is FI-cotorsion, (2) P ⊗ M is FI-cotorsion for every projective R-module P.
Proof. 1) ⇒ 2) Let F be a FI-flat R-module and P a projective R-module. As P projective, there exists a projective module P such that R I ∼ = P ⊕ P for some index set I. Now we have Next proposition gives a characterization of FI-cotorsion modules over a coherent ring. Proof. 1) ⇒ 2) For any FI-flat R-module N there exists an exact sequence of R-modules 0 → Now, applying the long exact sequence of the functor(Hom R (., M)), we get : By [7,Proposition 2.11] N ⊗ F is FI-flat since N is FI-flat and R is coherent, so On the other hand, applying the long exact sequence of the functor Hom R (., Hom R (F, M)) we get: projective. From [10, Theorem 2.75], Hom R (P ⊗ R F, M) ∼ = Hom R (P, Hom R (F, M)) and Hom R (K, Hom R (F, D)) ∼ = Hom R (K, Hom R (F, M)), then Ext 1 R (N, Hom R (F, M)) = 0 and so 3) ⇒ 1) Suppose that Hom R (P, M) is FI-cotorsion for every projective R-module P, then for In the following proposition, we show that FI-cotorsion modules are stable under direct product and summand. Proof. Follows from [10,Theorem 7.22 In the following proposition, we show how FI-cotorsion behave in a short exact sequence. Proof. Applying the long exact sequence of the functor Hom R (F, .), where F is FI-flat, we get: and the result holds.
In the following theorem we see when the character module M + of a module M is FIcotorsion.
Proof. Suppose that FP − id(M) = m for some non-negative integer m and consider the FP- Then we obtain the following short exact sequences: Applying the long exact sequence of the functor (. ⊗ R F) where F is a FI-flat module, we get: where every ends vanish since every E i is FP-injective for any 0 ≤ i ≤ m and F is a FI-flat.
Recall that R is called a IF ring if every injective R-module is flat. Next proposition gives a characterization of IF-ring using FI-cotorsion modules.

Proposition 6. A ring R is an IF-ring if and only if every FI-cotorsion R-module is injective.
Proof. The necessity is follows from [8, Proposition 2.9] since every module is FI-flat over any IF-rings. For the sufficiency, let E be injective module, then by Theorem 2.1 E + is FI-cotorsion and so injective by hypothesis. hence E is flat and R is IF-ring. there is an FI-cotorsion R-module which is not injective.
In Remark 1 we see that every FI-cotorsion module is cotorsion, in the following propositions we discus when the converse holds. On the next theorem we see the ring over which every R-modules is FI-cotorsion.
Theorem 2.2. Let R be a ring, then the following conditions are equivalent.
Moreover, if R satisfy one of the previous condition, then R is a perfect ring.
Proof. 1) ⇒ 2) Let F be a FI-flat R-module, so by hypothesis for any R-module M we have Recall that every semisimple ring is von Neumann regular and the converse is not true in general. Next corollary shows the converse is hold when every R-module is FI-cotorsion. Corollary 1. Let R be a ring, the following are equivalent: (1) R is a semisimple ring; (2) R is a von Neumann regular ring every FI-flat module is projective; (3) R is a von Neumann regular ring and every R-module is FI-cotorsion.
2) ⇒ 1) As R is a von Neumann regular ring, every module is flat and by Theorem 2.2, R is perfect and every flat module is projective. Then R is semisimple.

FI-FLAT COVER AND FI-COTORSION ENVELOPE
We recall the following notations, F C is class of FI-cotorsion modules, F F is Proof. Consider an exact sequence 0 → F → F → F → 0 with F and F are FI-flat. Applying the long exact sequence of functor (N ⊗ R ·) where N is an FP-injective R-module, we obtain the exact sequence: Hence Tor R 1 (N, F ) = 0 and F is FI-flat, therefor, (F F , F C ) is an hereditary cotorsion theory as claimed.
The next proposition is ensured the existence of the FI-flat cover and FI-cotorsion envelope of R-modules over an IF-ring.

Proposition 10.
Over an IF-ring R, every R-module has a F F -cover and a F C -envelope.
Proof. As R is an IF-ring, every FI-cotorsion module is injective by Proposition 7. Hence every FI-cotorsion module is pure-injective. Now, by Proposition 9, ⊥ F C = F F and ( ⊥ F C ) ⊥ = F C . So, (F F , F C ) is a cotorsion theory generated by F C ⊆ PI where PI denoted the class of pure-injective modules. Hence, (F F , F C ) is complete, and hence perfect by [11,Theorem 2.8]. So F F is a cover class and F C is an envelope class over an IF-ring.
In the next corollary, we see that the cokernel of F C -envelope is FI-flat and the kernel of the F F -cover of R-module is FI-cotorsion.

FI-COTORSION DIMENSIONS OF MODULES AND RINGS
4.1. FI-cotorsion dimension of modules. In this section we will investigate the FI-cotorsion dimension of modules and we study its properties and we give its characterization. From the previous definition we obtain the following remarks.
The following proposition gives a characterization of FI-cotorsion dimension.
Proposition 11. Let R be a ring and let M be a module and n ≥ 0. Then the following conditions are equivalent: (1) FI-cd R (M) ≤ n.
(2) in f {m| there exists an exact sequence (3) The integer k such that M admits a minimal FI-cotorsion resolution. Equivalently, there exists an exact sequence 0 Proof. (1) = (2) Let (1) = h, l = (2), suppose that h = l and suppose with out loss of generality that h > l. From the exact sequence 0 → M → M 0 → M 1 → · · · → M l−1 → M l → 0 such that every M i ∈ F C for 0 ≤ i ≤ l, we get the following short exact sequences: Applying the long exact sequence of the functor Hom R (F, .), we get: where every ends vanish since every M i is FI-cotorsion for every 0 ≤ i ≤ l and F is a FI-flat.
From the last sequence we have Ext 2  is closed under extensions. By Proposition 11 (5), N h is FI-cotorsion using the exact sequence   The first and the last term are zero since B is FI-cotorsion and F is FI-flat so: From Definition 6 FI-cd R (A) = FI-cd R (C) + 1.

Now, if
A is FI-cotorsion and for n = 1 in ( * ) we obtain: Corollary 3. Let {M i } i∈I be a family of modules.
Proof. Follows from [10, Theorem 7.14] since Ext n R (F, ∏ i∈I M i ) ∼ = ∏ i∈I Ext n R (F, M i ). Then we can deduce the result using Proposition 11.    For any ring R we have: The following result gives a characterization of the FI-cotorsion dimension of rings.
Proposition 14. Let R be a ring and n ≥ 0 an integer. Then the following conditions are equivalent: (1) FI-Cdim(R) ≤ n; Proof. The proof is obvious it follows from the definition and Proposition 11.
Proposition 15. Let R be a ring.
Proof. 1) Follows from the definition and Proposition 14.
2) Let M be a FI-flat module. As FI-Cdim(R) < ∞, we can suppose that pd(M) = n < ∞ by In the following theorem we characterize rings of FI-Cdim(R) ≤ 1. The first term and the last term vanish since M is FI-cotorsion and by hypothesis hence