American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) ISSN (Print) 2313-4410, ISSN (Online) 2313-4402 ยฉ Global Society of Scientific Research and Researchers http://asrjetsjournal.org/ Fp-Projective and Fp-Cotorsion Modules Shhaied Riad * Department of University Requirements-Faculty of Science and Arts, International University for Science and Technology, Damascus, Syria E-mail: rshhaied@yahoo.com Abstract Let R be a ring. The concepts of fp-projective and fp-cotorsion R-modules are defined. These modules together with the concept of the cotorsion theory are used to characterize left Noetherian rings, left perfect and right coherent rings, left coherent rings and left fp-regular rings. Moreover, some known characterizations of von Neumann regular rings are found. Keywords: Cotorsion theory; fp-projective modules; fp-cotorsion modules; Noetherian rings; perfect rings; coherent rings; fp-regular rings. 1. Introduction Throughout this paper, R is an associative ring with identity and all modules considered are unitary. For any R- module M, E(M) is the injective envelope of M, the character module Homโ„ค(M, โ„š/โ„ค) is denoted by M +. R-Mod (Mod-R) denotes the category of all left (right) R-modules. Hom(M, N) (resp., Ext1(M, N), Tor1(M, N)) means HomR(M, N) (resp., ExtR1 (M, N), Tor1R(M, N)). Recall that a left R-module M is FP-injective [1], if Ext1(A, M) = 0 for every finitely presented left R-module A, M is F-injective [2] if Ext1(R/I, M) = 0 for every finitely generated left ideal I, M is fp-injective if Ext1(R/I, M) = 0 for every finitely presented left ideal I. A right R-module M is fp-flat, if Tor1(M, R/I) = 0 for every finitely presented left ideal I. fp-injective (or almost F-injective) and fp-flat (or almost F-flat) modules have been introduced and studded in [3]. ------------------------------------------------------------------------ * Corresponding author. 132 http://asrjetsjournal.org/ American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 17, No 1, pp 132-143 Let M be a left R-module. Then M is FP-projective (F-projective), if Ext1(M, N) =0 for any FP-injective (F- injective) left R-module N, M is cotorsion if Ext1(N, M) = 0 for any flat left R-module N. More information for the concepts of FP-projective, F-projective and cotorsion modules can be found in [4], [5] and [6]. Given a class ๐’ž๐’ž โŠ† Mod-R, then ๐’ž๐’žโŠฅ = {N โˆˆ Mod-R | Ext1(C, N) = 0 for all C โˆˆ ๐’ž๐’ž}, โŠฅ๐’ž๐’ž = {N โˆˆ Mod-R | Ext1 (N, C) = 0 for all C โˆˆ ๐’ž๐’ž}. When ๐’ž๐’ž โŠ† R-Mod, ๐’ž๐’žโŠฅ and โŠฅ๐’ž๐’ž can be defined similarly. Let ๐’ž๐’ž be a class of left (right) R-modules and M be a left (right) R-module. A homomorphism ฯ†: M โ†’ F where F โˆˆ ๐’ž๐’ž is called a ๐’ž๐’ž-preenvelope [7] of M if for any homomorphism f: M โ†’ Fโ€ฒ with Fโ€ฒ โˆˆ ๐’ž๐’ž, there is a homomorphism g: F โ†’ Fโ€ฒ such that g ฯ† = f. If the only such g are automorphisms of F when Fโ€ฒ = F and f = ฯ†, then the ๐’ž๐’ž-preenvelope ฯ† is called a ๐’ž๐’ž-envelope of M. Following [7, Definition 7.1.6], a monomorphism ฮฑ: M โ†’ C with C โˆˆ ๐’ž๐’ž is said to be a special ๐’ž๐’ž-preenvelope of M if coker (ฮฑ) โˆˆ โŠฅ๐’ž๐’ž. Dually we have the definitions of ๐’ž๐’ž-precover, special ๐’ž๐’ž-precover and -cover. ๐’ž๐’ž-envelopes (๐’ž๐’ž-covers) may not exist in general, but if they exist, they are unique up to isomorphism. Let ๐’œ๐’œ, โ„ฌ โŠ† Mod-R. The pair (๐’œ๐’œ, โ„ฌ) is called a cotorsion theory [7], if ๐’œ๐’œโŠฅ= โ„ฌ and โŠฅโ„ฌ = ๐’œ๐’œ. The cotorsion theory (๐’œ๐’œ, โ„ฌ), when ๐’œ๐’œ, โ„ฌ โŠ† R-Mod can be defined similarly. A cotorsion theory (๐’œ๐’œ, โ„ฌ) is generated (cogenerated) by a class ๐’ž๐’ž โŠ† Mod-R, if ๐’œ๐’œ = โŠฅ๐’ž๐’ž (โ„ฌ = ๐’ž๐’žโŠฅ). A cotorsion theory (๐’œ๐’œ, โ„ฌ) is said to be perfect (complete) if every right R-module has a โ„ฌ-envelope and an ๐’œ๐’œ-cover (a special โ„ฌ-preenvelope and a special ๐’œ๐’œ-precover) (see [8] and [9]). A cotorsion theory (๐’œ๐’œ, โ„ฌ) is said to be hereditary [8] if whenever 0 โ†’ Kโ†’ P โ†’ F โ†’ 0 is exact with P, F โˆˆ ๐’œ๐’œ, then K is also in ๐’œ๐’œ. By [8, Proposition 1.2], (๐’œ๐’œ, โ„ฌ) is hereditary if and only if 0 โ†’ Aโ†’ B โ†’C โ†’ 0 is exact with A, B โˆˆ โ„ฌ, then C is also in โ„ฌ. In section two of this note, we first extend the concepts of F-projective modules and cotorsion modules to fp- projective modules and fp-cotorsion modules respectively, and then give some characterizations and properties of fp-injective modules and fp-projective modules. We also prove that (fp-proj, fp-inj) is a complete cotorsion theory, and (fp-fl, fp-cot) is a perfect cotorsion theory, where fp-proj (fp-inj) denotes the class of all fp- projective (fp-injective) modules, and fp-fl ( fp-cot) the class of all fp-flat (fp-cotorsion ) modules. The activity of section two will be appeared in section three, where we present some characterizations of left Noetherian rings, left coherent and right perfect rings, left coherent rings, left fp-regular [3] rings by fp- projective and fp-cotorsion R-modules. Moreover, some known characterizations of von Neumann regular rings are found. 2. Definitions and first results We first introduce the concepts of fp-projective and fp-cotorsion modules. 133 American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 17, No 1, pp 132-143 2.1 Definition Let M be a left R-module. Then we call (1) M is fp-projective if Ext1(M, N) = 0 for any fp-injective left R-module N. (2) M is fp-cotorsion if Ext1(N, M) = 0 for any fp-flat left R-module N. The right version can be defined similarly. For any ring R, it is clear that every projective left R-module is fp-projective, R/I is fp-projective left R-module for every finitely presented left ideal I, every fp-projective left R-module is F-projective and FP-projective, every injective left R-module is fp-cotorsion and every fp-cotorsion left R-module is cotorsion. Next, we give some characterizations of fp-injective R-modules by fp-projective R-modules. 2.2 Theorem Let M be a left R-module. The following statements are equivalent. (1) M is fp-injective left R-module. (2) M is injective with respect to every exact sequence 0 โ†’ C โ†’ B โ†’A โ†’ 0 of left R-modules with A fp- projective. (3) M is injective with respect to every exact sequence 0 โ†’ K โ†’ P โ†’A โ†’ 0 of left R-modules with P projective and A fp-projective. (4) Every fp-projective R-module is projective with respect to every exact sequence of left R-modules 0 โ†’ M โ†’ M' โ†’ M" โ†’ 0. (5) Every fp-projective left R-module is projective with respect to the canonical exact sequence 0 โ†’ M โ†’ E(M) โ†’ E(M)โˆ•M โ†’ 0. (6) There exists an exact sequence 0 โ†’ M โ†’ M' โ†’ M" โ†’ 0 of left R-modules with M' injective, such that every fp-projective left R-module is projective with respect to this sequence. (7) There exists an exact sequence 0 โ†’ M โ†’ M' โ†’ M" โ†’ 0 of left R-modules with M' fp-injective, such that every fp-projective left R-module is projective with respect to this sequence. Proof. (1) โ‡’ (2) โ‡’ (3) are clear. (3) โ‡’ (1). Let I be a finitely presented left ideal of R. We have the exact sequence 0 โ†’ I โ†’ R โ†’ R/I โ†’ 0, which induces the exact sequence Hom(R, M) โ†’ Hom(I, M) โ†’ Ext1(R/I, M) โ†’ 0. Since R/I is fp-projective, Hom(R, M) โ†’ Hom(I, M) โ†’0 is exact by (3). So Ext1(R/I, M) = 0 and so (1) follows. (1) โ‡’ (4). Let M be an fp-injective left R-module. Then the sequence Hom(P, M') โ†’ Hom(P, M")โ†’ Ext1(P, M) = 0 is exact for every fp-projective left R-module P, and so (4) follows. 134 American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 17, No 1, pp 132-143 (4) โ‡’ (5) โ‡’ (6) โ‡’ (7) are obvious. (7) โ‡’ (1). Assume that 0 โ†’ M โ†’ M' โ†’ M" โ†’ 0 is exact sequence of left R-modules with M' fp-injective and, I is any finitely presented left ideal of R. Since R/I is fp-projective left R-module, then by (7), the sequences Hom(R/I, M') โ†’ Hom(R/I, M") โ†’ Ext1(R/I, M) โ†’ Ext1(R/I, M' ) = 0 and Hom(R/I, M') โ†’ Hom(R/I, M") โ†’ 0 are exact. This implies that Ext1(R/I, M) = 0, and (1) follows. โ–ก Now, we give some characterizations of fp-projective R-modules by fp-injective R-modules. 2.3 Theorem Let M be a left R-module. The following statements are equivalent. (1) M is left fp-projective. (2) M is projective with respect to every exact sequence 0 โ†’ A โ†’ B โ†’ C โ†’ 0 of left R-modules with A fp-injective. (3) M is projective with respect to every exact sequence 0 โ†’ A โ†’ F โ†’ K โ†’ 0 of R-modules with F injective and A fp-injective. (4) Every fp-injective R-module is injective with respect to every exact sequence of left R-modules 0 โ†’ M" โ†’ M' โ†’ M โ†’ 0. (5) There exists an exact sequence 0 โ†’ M" โ†’ M' โ†’ M โ†’ 0 of left R-modules with M' projective, such that every fp-injective left R-module is injective with respect to this sequence. (6) There exists an exact sequence 0 โ†’ M" โ†’ M' โ†’ M โ†’ 0 of left R-modules with M' fp-projective, such that every fp-injective left R-module is injective with respect to this sequence. Proof. (1) โ‡’ (2). Let M be an fp-projective left R-module. Then the sequence Hom(M, B) โ†’ Hom(M, C) โ†’ Ext1(M, A) = 0 is exact for every fp-injective R-module A. Hence (2) holds. (2) โ‡’ (3) is clear. (3) โ‡’ (1). Let 0 โ†’ A โ†’ F โ†’ K โ†’ 0 be an exact sequence of left R-modules with F injective and A fp- injective. Then the sequence Hom(M, F) โ†’ Hom(M, K) โ†’ 0 is exact by (3). So Ext1(M, A) = 0 for any fp- injective left R-module A. Hence M is fp-projective left R-module. (1) โ‡’ (4). Let M be an fp-projective left R-module, F be an fp-injective left R-module. Then the sequence Hom(M', F) โ†’ Hom(M", F)โ†’ Ext1(M, F) = 0 is exact. So (4) holds. (4) โ‡’ (5) โ‡’ (6) are obvious. (6) โ‡’ (1). Let 0 โ†’ M" โ†’ M' โ†’ M โ†’ 0 be an exact sequence of left R-modules with M' fp-projective. Then for any fp-injective left R-module N, the sequence Hom(M', N) โ†’ Hom(M", N) โ†’ Ext1(M, N) โ†’ Ext1(M', N ) = 0 is exact, and the sequence (M', N) โ†’ Hom(M", N) โ†’ 0 is exact by (6).This implies that Ext1(M, N) = 0, and (1) 135 American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 17, No 1, pp 132-143 follows. โ–ก A cotorsion theory (๐’œ๐’œ, โ„ฌ) is said to have enough injectives (resp., enough projectives) (see. [10, p. 5]) if for every module L there is a short exact sequence 0 โ†’ L โ†’B โ†’ A โ†’ 0 (resp., 0 โ†’ B โ†’ A โ†’ L โ†’ 0) such that B โˆˆ โ„ฌ and A โˆˆ ๐’œ๐’œ. 2.4 Lemma [10, Theorem 10]. Every cotorsion theory which is cogenerated by a set of modules has enough projectives and enough injectives. โ–ก 2.5 Lemma (see [7, p.159]). If a cotorsion theory (๐’œ๐’œ, โ„ฌ) has enough projectives and enough injectives, then every module has a special โ„ฌ-preenvelope and a special ๐’œ๐’œ-precover. โ–ก In what follows, we assume that fp-proj (fp-inj) denotes the class of all fp-projective (fp-injective) left R- modules, and fp-fl (fp-cot) denotes the class of all fp-flat (fp-cotorsion ) right R-modules. 2.6 Theorem Let R be a ring. Then (fp-proj, fp-inj) is a complete cotorsion theory. Proof. Let ๐’ž๐’ž ={R/I | I is finitely presented left ideal I of R}. Then ๐’ž๐’žโŠฅ={N โˆˆ R-Mod | Ext1(R/I, N)} = 0 for any finitely presented left ideal I of R}, so ๐’ž๐’žโŠฅ = fp-inj. On the other hand, we have โŠฅ( ๐’ž๐’žโŠฅ) = {N โˆˆ R-Mod | Ext1(N, C) = 0 for all C โˆˆ ๐’ž๐’žโŠฅ}, so โŠฅ( ๐’ž๐’žโŠฅ) = fp-proj. Now we have ๐’ž๐’žโŠฅ = ( โŠฅ( ๐’ž๐’žโŠฅ))โŠฅ. It follows that the pair (fp-proj, fp- inj) is a cotorsion theory that is cogenerated by the class ๐’ž๐’ž. So the result follows from Lemma 2.4 and Lemma 2.5 โ–ก Following [9], for a class of (right, resp. left) R-modules ๐’ž๐’ž,we put ๐’ž๐’žั‚ = {N โˆˆ R-Mod | Tor1(C, N) = 0 for all C โˆˆ ๐’ž๐’ž}, resp. ั‚๐’ž๐’ž = {N โˆˆ Mod-R | Tor1(N, C) = 0 for all C โˆˆ ๐’ž๐’ž}. Let ๐’œ๐’œ โŠ† Mod-R, โ„ฌ โŠ† R-Mod. The pair (๐’œ๐’œ, โ„ฌ) is called a Tor-torsion theory if ๐’œ๐’œ = ั‚โ„ฌ and โ„ฌ =๐’œ๐’œั‚. In what follows, by ๐’ซ๐’ซโ„ we denote the class of all pure injective right R-modules. 2.7 Lemma [see. 9, Lemma 1.11]. Let ๐’œ๐’œ โŠ† Mod-R, โ„ฌ โŠ† R-Mod and (๐’œ๐’œ, โ„ฌ) be a Tor-torsion theory. Then (๐’œ๐’œ, ๐’œ๐’œโŠฅ) is a cotorsion theory generated by ๐’ž๐’ž = {M + | M โˆˆ โ„ฌ} โŠ† ๐’ซ๐’ซโ„. โ–ก 136 American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 17, No 1, pp 132-143 2.8 Lemma [see.9, Theorem 2.8]. Let R be a ring and (๐’œ๐’œ, โ„ฌ) be a cotorsion theory generated by a class ๐’ž๐’ž โŠ† ๐’ซ๐’ซโ„. Then (๐’œ๐’œ, โ„ฌ) is complete. Moreover, ๐’œ๐’œ is a cover class and โ„ฌ is an envelope class. โ–ก 2.9 Theorem Let R be a ring. Then (fp-fl, fp-cot) is a perfect cotorsion theory. Proof. Let โ„ฌ ={R/I | I is finitely presented left ideal I of R}, ั‚โ„ฌ = {N โˆˆ Mod-R | Tor1(N, R/I) = 0 for any finitely presented left ideal I of R}. Let ๐’œ๐’œ = fp-fl. Then we have ๐’œ๐’œั‚ = {N โˆˆ R-Mod | Tor1(M, N) = 0 for any fp-flat right R-module M}. Since M is a fp-flat right R-module if and only if Tor1(M, R/I) = 0 for every finitely presented left ideal I, ๐’œ๐’œ = ั‚โ„ฌ and โ„ฌ = ๐’œ๐’œั‚. So (๐’œ๐’œ, โ„ฌ) is a Tor-torsion theory. By lemma 2.7, (๐’œ๐’œ, ๐’œ๐’œโŠฅ) is a cotorsion theory generated by ๐’ž๐’ž = {M + | M โˆˆ โ„ฌ} โŠ† ๐’ซ๐’ซโ„. By lemma 2.8, (๐’œ๐’œ, ๐’œ๐’œโŠฅ) is a perfect cotorsion theory. Finally we note that ๐’œ๐’œโŠฅ = {N โˆˆ Mod-R | Ext1(M, N) = 0, for any fp-flat right R-module M} = fp-cot. Thus (fp- fl, fp-cot) is a perfect cotorsion theory, as desired. โ–ก 3. Characterizations of some classes of rings It is well known that a ring R is left Noetherian if and only if every F-injective left R-module is injective [11, Corollarly1.10]. Next, we characterize left Noetherian rings by fp-injective left R-modules and fp-projective left R-modules. 3.1 Theorem For any ring R, the following statements are equivalent. (1) R is left Noetherian. (2) Every left R-module is fp-projective. (3) Every finitely generated left R-module is fp-projective. (4) Every cyclic left R-module is fp-projective. (5) Every fp-injective left R-module is injective. (6) (fp-proj, fp-inj) is hereditary, and every fp-injective left R-module is fp-projective. Proof. (2) โ‡’ (3) โ‡’ (4) are clear. (4) โ‡’ (5). Let M any fp-injective left R-module and I any left ideal of R. Then Ext1(R/I, M) = 0 by (4).Thus M is injective, as required. (5) โ‡’ (2). Let M any left R-module, 0 โ†’ A โ†’ B โ†’ C โ†’ 0 be an exact sequence with A fp-injective. Since A is injective by (5), the sequence Hom(M, B) โ†’ Hom(M, C) โ†’ Ext1(M, A) = 0 is exact. So M is fp-projective by Theorem 2.3 (2). 137 American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 17, No 1, pp 132-143 (5) โ‡’ (1). It is clear that every F-injective left R-module is fp-injective. So by (5) and [11, Corollary1.10], R is left Noetherian. (1) โ‡’ (5). Let R be a left Noetherian ring. Then every left ideal of R is finitely presented. So every fp-injective left R-module is F-injective. Thus (5) holds by [11, Corollary1.10]. (2) โ‡’ (6) is clear. (6) โ‡’ (2). Let M any left R-module. Since (fp-proj, fp-inj) is a complete cotorsion theory by Theorem 2.6 , M has a special fp-injective preenvelope, i.e. there is a short exact sequence 0 โ†’ M โ†’ F โ†’ P, where F is fp- injective and P is fp-projective. Since P is fp-projective, and (fp-proj, fp-inj) is hereditary and F is fp-projective by (6), M must be fp-projective. Thus (2) holds. โ–ก Recall that a ring R is called left coherent if every finitely generated left ideal of R is finitely presented. Coherent rings and their generations have been studied and characterized by many authors (see. for example [1 ,3,5,6,12,13]). Next, we give some characterizations of left coherent rings in terms of fp-projective and fp-cotorsion left R- modules. 3.2 Theorem The following statements are equivalent for a ring R. (1) R is left coherent. (2) Every fp-injective left R-module is FP-injective. (3) Every fp-injective left R-module is F-injective. (4) Every fp-flat right R-module is flat. (5) Every FP-projective left R-module is fp-projective. (6) Every finitely presented left R-module is fp-projective. (7) Every F-projective left R-module is fp-projective. (8) For every finitely generated left ideal I of R, R/I is fp-projective. (9) Every cotorsion right R-module is fp-cotorsion. Proof. (1) โ‡” (2) โ‡” (3) โ‡” (4) by [3, theorem 2.8]. (5) โ‡’ (6) is clear, since every finitely presented left R-module is FP-projective. (6) โ‡’ (2). Let M be an fp-injective left R-module and A any finitely presented left R-module. Then Ext1(A, M) = 0 by (6).Thus M is FP-injective. (2) โ‡’ (5). Let M be an FP-projective left R-module and N any fp-injective left R-module. Since N is FP- 138 American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 17, No 1, pp 132-143 injective by (2), Ext1(M, N) = 0. Thus M is fp-projective. (3) โ‡’ (7). Let M be an F-projective left R-module and N any fp-injective left R-module. Since N is F-injective by (3), Ext1(M, N) = 0. Thus M is fp-projective. (7)โ‡’(8) since R/I is F-projective left R-module for any finitely generated left ideal I of R. (8) โ‡’ (3). Let M be an fp-injective left R-module and I any finitely generated left ideal. Then Ext1(R/I, M) = 0 by (8). Thus M is F-injective. (4) โ‡’ (9). Let M be a cotorsion right R-module and N any fp-flat right R-module. Since N is flat by (4), Ext1(N, M) = 0. So M is fp-cotorsion. (9) โ‡’ (4) is similar to (4) โ‡’ (9). โ–ก Recall that a ring R is called right perfect if every left R-module has a projective cover [14]. A classical theorem of Bass states that R is right perfect if and only if every flat right R-module is projective [15, Theorem and Definition 2.1]. Using this fact with the Theorem 3.2, we get the following characterizations of left coherent and right perfect ring in terms of fp-injective and fp-cotorsion modules. 3.3 Theorem The following statements are equivalent for a ring R. (1) R is left coherent and right perfect. (2) Every right R-module M is fp-cotorsion. (3) (fp-fl, fp-cot) is hereditary and every fp-flat right R-module is fp-cotorsion . Proof. (1) โ‡’ (2). Let M be a right R-module and N any fp-flat right R-module. Since R is left coherent, N is flat right R-module by [3, Theorem 2.8 (4)]. Since R is right perfect, N is projective right R-module, so Ext1(N, M) = 0. Hence M is fp-cotorsion. (2) โ‡’ (1). Since every cotorsion right R-module is fp-cotorsion by (2), R is left coherent by Theorem 3.2 (9). On the other hand, if N is a flat right R-module, then for any right R-module M, Ext1(N, M) = 0 by (2). So N is projective right R-module and hence R is right perfect. (2) โ‡’ (3) is clear. (3) โ‡’ (2). Let M be any right R-module. Since (fp-fl, fp-cot) is a perfect cotorsion theory, M has an fp-flat cover by Theorem 2.9, i.e. there exists an exact sequence 0 โ†’ L โ†’ F โ†’ M โ†’ 0 with F fp-flat right R- module and L fp-cotorsion right R-module. By (3) F is fp-cotorsion right R-module, and so M must be fp- cotorsion. Thus (2) holds. โ–ก 139 American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 17, No 1, pp 132-143 It is well known that a ring R is a von Neumann regular ring if and only if every finitely generated left ideal of R is a direct summand of R. Regular rings and their generations have been studied and characterized by many authors (see. for example [3,5,6,8,16]). Following [3, p. 124], we call a ring R, a left fp-regular (or left almost regular), if every finitely presented left ideal of R is a direct summand of R. The right fp-regular ring can be defined similarly. Next, we give some characterizations of left fp-regular rings in terms of fp-projective and fp-cotorsion R-modules. 3.4 Theorem For any ring R, the following statements are equivalent. (1) R is a left fp-regular. (2) For any finitely presented left ideal I, R/I is projective left R-module. (3) Every left R-module is fp-injective. (4) Every right R-module is fp-flat. (5) Every fp-projective left R-module is projective. (6) Every cotorsion left R-module is fp-injective. (7) Every fp-cotorsion right R-module is injective. (8) Every cotorsion right R-module is fp-flat. (9) (fp-fl, fp-cot) is hereditary and every fp-cotorsion right R-module is fp-flat. (10) (fp-proj, fp-inj) is hereditary and every fp-projective left R-module is fp- injective. Proof. (1) โ‡” (2) โ‡” (3) โ‡” (4) by [3, Theorem 3.5]. (3) โ‡’ (5). Let M any fp-projective left R-module. Then by (3), Ext1(M, N) = 0 for any left R-module N, so M is projective. (5) โ‡’ (2) is clear, since R/I is fp-projective left R-module for every finitely presented left ideal I. (3) โ‡’ (6) is clear. (6) โ‡’ (2). Let I any finitely presented left ideal of R, M any cotorsion left R-module. We have M is fp-injective by (6), and R/I is fp-projective left R-module. So M is injective with respect to the exact sequence 0 โ†’ I โ†’ R โ†’ R/I โ†’ 0 of left R-modules by Theorem 2.2 (2), and so Ext1(R/I, M) = 0. By the arbitrariness of the cotorsion left R-module M, it follows that R/I is flat. Thus R/I is projective since R/I is finitely presented. (4) โ‡’ (7). Let M be an fp-cotorsion right R-module. Then, by (4) Ext1(N, M) = 0 for any right R-module N, so M is injective right R-module. (7) โ‡’ (4). Let N be a right R-module, M any fp-cotorsion right R-module. Then, by (7) Ext1(N, M) = 0, so N is fp-flat right R-module. 140 American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 17, No 1, pp 132-143 (4) โ‡’ (8) is clear. (8) โ‡’ (3). Let M be any left R-module. Then M + is pure-injective right R-module and hence M + is a cotorsion right R-module. So M + is fp-flat right R-module by (8), and so M is fp-injective left R-module by [3, Theorem 2.7 (1)]. (4) โ‡’ (9) and (3) โ‡’ (10) are obvious. (9) โ‡’ (4) Let M be any right R-module. By Theorem 2.9 there exists an exact sequence 0 โ†’ M โ†’ C โ†’ L โ†’ 0 with C fp-cotorsion right R-module and L fp-flat right R-module. So C is fp-flat right R-module by (9). Since (fp-fl, fp-cot) is hereditary M is fp-flat right R-module. (10) โ‡’ (3). Let M be any left R-module. By Theorem 2.6 there exists an exact sequence 0 โ†’ N โ†’ P โ†’ M โ†’ 0, where P is fp- projective left R-module and N is fp-injective left R-module. So P is fp-injective by (10).Since (fp-proj, fp-inj) is hereditary, M is fp-injective. โ–ก 4. Conclusion It is not known whether, the left fp-regular right coherent ring is von Neumann regular ring. On the other hand , it is clear that every von Neumann regular ring is right and left fp-regular ring, but the converse is not true as shown by [3, Example 3.6]. By Theorem 3.2(4) and Theorem 3.4(4), it is obvious that R is a von Neumann regular ring if and only if R is a left fp-regular left coherent. Using this fact with Theorem 3.2 and Theorem 3.4, we can get the following known characterizations of von Neumann regular ring: (1) R is von Neumann regular ring. (2) Every right R-module is flat. (3) every left R-module is F-injective. (4) every cotorsion left R-module is F-injective. (5) every cotorsion right R-module is flat. (6) Every F-projective left R-module is projective. (7) (F-proj, F-inj) is hereditary and every F-projective left R-module is F-injective. โ–ก Note that the equivalences (1) โ‡” (2) โ‡” (3) have been proved by other ways in [11, Theorem 1.11], (1) โ‡” (4) โ‡” (5) in [17, Theorem 3.2.2]. and (1) โ‡” (6) โ‡” (7) in [6, Theorem 3.2]. 5. Recommendations It is well known that a ring R is quasi-Frobenius when every left R-module embeds in a free left R-module. If the embedding is restricted to finitely generated left R-modules, then R is called left FGF ring, and the question of whether any left FGF ring is QF is known as the FGF conjecture (or Faithโ€™s Problem). This conjecture is still 141 American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 17, No 1, pp 132-143 an open question. But there are many affirmative answers to this question with different additional conditions. (see [18]โ€“[20]). It is recommended to characterize the quasi-Frobenius rings in terms of fp-projective and fp- cotorsion left R-modules, and then reach a new affirmative answers to the FGF conjecture with additional condition. Acknowledgment The author would like to thank the referee for the valuable comments and suggestions. References [1] B. Stenstrรถm, Coherent rings and FP-injective modules, J. London Math. Soc., 2, 323-329, 1970. [2] R. N. Gupta, On F-injective modules and semihereditary rings, Proc. Nat. Inst. Sci.India, Part A, 35, 323-328,1969. [3] Zhu Zhanmin, Almost F-injective modules and almost flat modules, International Electronic Journal of Algebra, Volume 16,115-126, 2014. 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Damiano, coflat rings and modules, Pacific Journal of Mathematics, Vol. 81, No. 2, 349- 369, 1979. [12] X. X. Zhang, J. L. Chen and J. Zhang, On (m, n)-injective modules and (m, n)-coherent rings, Algebra Colloq., 12 ,149-160, 2005. [13] J. L. Chen and N. Q. Ding, On n-coherent rings, Comm. Algebra, 24, 3211-3216, 1996. [14] H. Bass, โ€œFinitistic dimension and a homological generalization of semiprimary, rings,โ€ Trans. Amer. Math. Soc. 95, 466โ€“488, 1960. [15] Carl Faith, When Self- injective Rings are QF:A Report on a Problem, Journal of Algebra and Its Applications,Vol. 1, No. 1, 75โ€“105, 2002. [16] MAO Li-xin, DING Nan-qing, Notes on cotorsion modules [J]. Comm. Algebra, 33(1): 349โ€“360, 2005. [17] Xu, J. (1996). Flat Covers of Modules. Lecture Notes in Math. Vol. 1634. Berlin-Heidelberg-New York: Springer Verlag. 142 American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 17, No 1, pp 132-143 [18] Faith C, Huynh DV, When self-injective rings are QF: a report on a problem. J. Algebra Appl. 1, 75- 105, 2002. [19] Nicholson, WK, Yousif, MF, Quasi-Frobenius Rings. Cambridge University Press, Cambridge, 35- 163. [20] Shen, L, Chen, JL, , New characterizations of quasi-Frobenius rings. Commun. Algebra 34, 2157- 2165, 2006. 143