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Notes on FP-Projective Modules and FP-Injective Modules

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Notes on FP-Projective Modules and FP-Injective Modules February 17, 2006 17:49 Proceedings Trim Size: 9in x 6in mao-ding NOTES ON FP -PROJECTIVE MODULES AND FP -INJECTIVE MODULES LIXIN MAO Department of Mathematics, Nanjing Institute of Technology Nanjing 210013, P.R. China Department of Mathematics, Nanjing University Nanjing 210093, P.R. China E-mail: [email protected] NANQING DING Department of Mathematics, Nanjing University Nanjing 210093, P.R. China E-mail: [email protected] In this paper, we study the FP -projective dimension under changes of rings, es- pecially under (almost) excellent extensions of rings. Some descriptions of FP - injective envelopes are also given. 1. Introduction Throughout this paper, all rings are associative with identity and all mod- ules are unitary. We write MR (RM) to indicate a right (left) R-module, and freely use the terminology and notations of [1, 4, 9]. 1 A right R-module M is called FP -injective [11] if ExtR(N; M) = 0 for all ¯nitely presented right R-modules N. The concepts of FP -projective dimensions of modules and rings were introduced and studied in [5]. For a right R-module M, the FP -projective dimension fpdR(M) of M is de¯ned to be the smallest integer n ¸ 0 such n+1 that ExtR (M; N) = 0 for any FP -injective right R-module N. If no such n exists, set fpdR(M) = 1. M is called FP -projective if fpdR(M) = 0. We note that the concept of FP -projective modules coincides with that of ¯nitely covered modules introduced by J. Trlifaj (see [12, De¯nition 3.3 and Theorem 3.4]). It is clear that fpdR(M) measures how far away a right R-module M is from being FP -projective. The right FP -projective dimension rfpD(R) of a ring R is de¯ned as supffpdR(M): M is a ¯nitely 1 February 17, 2006 17:49 Proceedings Trim Size: 9in x 6in mao-ding 2 generated right R-moduleg and measures how far away a ring R is from being right noetherian (see [5, Proposition 2.6]). Let C be a class of right R-modules and M a right R-module. A ho- momorphism Á : M ! F with F 2 C is called a C-preenvelope of M [4] 0 0 if for any homomorphism f: M ! F with F 2 C, there is a homo- 0 morphism g : F ! F such that gÁ = f. Moreover, if the only such g 0 are automorphisms of F when F = F and f = Á, the C-preenvelope Á is called a C-envelope of M.A C-envelope Á : M ! F is said to have 0 the unique mapping property [3] if for any homomorphism f: M ! F 0 0 with F 2 C, there is a unique homomorphism g : F ! F such that gÁ = f. Following [4, De¯nition 7.1.6], a monomorphism ® : M ! C with C 2 C is said to be a special C-preenvelope of M if coker(®) 2 ?C, where ? 1 C = fF : ExtR(F; C) = 0 for all C 2 Cg. Dually we have the de¯nitions of a (special) C-precover and a C-cover (with unique mapping property). Spe- cial C-preenvelopes (resp., special C-precovers) are obviously C-preenvelopes (resp., C-precovers). Denote by FPR (resp., FIR) the class of FP -projective (resp., FP - injective) right R-modules. In what follows, special FPR-(pre)covers (resp., FIR-(pre)envelopes) will be called special FP -projective (pre)covers (resp., FP -injective (pre)envelopes). We note that (FPR, FIR) is a cotorsion theory (for the category of right R-modules) which is cogenerated by the representative set of all ¯nitely presented right R-modules (cf. [4, De¯nition 7.1.2]). Thus, by [4, Theorem 7.4.1 and De¯nition 7.1.5], every right R-module M has a special FP - injective preenvelope, i.e., there is an exact sequence 0 ! M ! F ! L ! 0, where F 2 FIR and L 2 FPR; and every right R-module has a special FP - projective precover, i.e., there is an exact sequence 0 ! K ! F ! M ! 0, where F 2 FPR and K 2 FIR. We observe that, if ® : M ! F is an FP - injective envelope of M, then coker(®) is FP -projective, and if ¯ : F ! M is an FP -projective cover of M, then ker(¯) is FP -injective by Wakamatsu's Lemmas [4, Propositions 7.2.3 and 7.2.4]. A ring S is said to be an almost excellent extension of a ring R [14, 15] if the following conditions are satis¯ed: (1) S is a ¯nite normalizing extension of a ring R [10], that is, R and S have the same identity and there are elements s1; ¢ ¢ ¢ ; sn 2 S such that S = Rs1 + ¢ ¢ ¢ + Rsn and Rsi = siR for all i = 1; ¢ ¢ ¢ ; n. (2) RS is flat and SR is projective. (3) S is right R-projective, that is, if MS is a submodule of NS and MR February 17, 2006 17:49 Proceedings Trim Size: 9in x 6in mao-ding 3 is a direct summand of NR, then MS is a direct summand of NS. Further, S is an excellent extension of R if S is an almost excellent extension of R and S is free with basis s1; ¢ ¢ ¢ ; sn as both a right and a left R-module with s1 = 1R. The concept of excellent extension was introduced by Passman [7] and named by Bonami [2]. The notion of almost excellent extensions was introduced and studied in [14, 15] as a non-trivial generalization of excellent extensions. In this paper, we ¯rst study the FP -projective dimension under changes of rings. Let R and S be right coherent rings (i.e., rings such that every ¯nitely generated right ideal is ¯nitely presented) and ' : R ! S be a surjective ring homomorphism with S projective as a right R-module and flat as a left R-module. It is proven that fpdS(M) = fpdR(M) for any right S-module MS, and hence rfpD(S) · rfpD(R). Let S be a ¯nite normalizing extension (in particular, an (almost) ex- cellent extension) of a ring R. It is well known that R is right noetherian if and only if S is right noetherian [8, Proposition 5]. It seems natural to generalize descent of right noetherianess to right FP -projective dimen- sions in the case when S is an (almost) excellent extension of a ring R. We show that if R and S are right coherent rings and S is an almost ex- cellent extension of R, then fpdR(M) = fpdS(M) for any right S-module MS, and rfpD(S) · rfpD(R), the equality holds if rfpD(R) < 1. We also show that, for a right coherent ring R, rfpD(R) · 2 and every (resp. FP -injective) right R-module has an FP -projective envelope if and only if every (resp. FP -injective) right R-module has an FP -projective envelope with the unique mapping property. Although the class of FP -injective R-modules is not enveloping (a class C is enveloping if every R-module has a C-envelope) (see [12, Theorem 4.9]), an individual R-module may have FP -injective envelopes. Some descrip- tions of an FP -injective envelope of an R-module are given. For example, it is shown that, if MR has an FP -injective envelope and is a submodule of an FP -injective right R-module L, then the inclusion i : M ! L is an FP -injective envelope of M if and only if L=M is FP -projective and any endomorphism γ of L such that γi = i is a monomorphism if and only if L=M is FP -projective and there are no nonzero submodules N of L such that M \ N = 0 and L=(M © N) is FP -projective. It is also shown that if R is a right coherent ring and MR has an FP -projective cover, then MR has a special FP -injective preenvelope ® : M ! N such that N has an FP -projective cover. Finally we consider FP -projective precovers under February 17, 2006 17:49 Proceedings Trim Size: 9in x 6in mao-ding 4 almost excellent extensions of rings. Let S be an almost excellent exten- sion of a ring R, it is proven that if θ : NS ! MS is an S-epimorphism, then θ : NR ! MR is a special FP -projective precover of MR if and only if θ : NS ! MS is a special FP -projective precover of MS. 2. Results We start with Lemma 2.1. Let ' : R ! S be a surjective ring homomorphism with SR projective and MS a right S-module (and hence a right R-module). (1) If MS is ¯nitely presented, then MR is ¯nitely presented. (2) If MS is FP -projective, then MR is FP -projective. Proof. (1). Since MS is ¯nitely presented, there is an exact sequence 0 ! K ! P ! M ! 0 of right S-modules with K ¯nitely generated and P ¯nitely generated projective. Since ' : R ! S is surjective, it is easy to see that K is a ¯nitely generated right R-module and P is a ¯nitely generated projective right R-module by [9, Theorem 9.32] (for SR is projective). Therefore M is a ¯nitely presented right R-module. (2). If MS is FP -projective, then MS is a direct summand in a right S- module N such that N is a union of a continuous chain, (N® : ® < ¸), for a cardinal ¸, N0 = 0, and N®+1=N® is a ¯nitely presented right S-module for all ® < ¸ (see [12, De¯nition 3.3]).
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