RESOLUTIONS AND DIMENSIONS OF RELATIVE INJECTIVE MODULES AND RELATIVE FLAT MODULES

Abstract

Abstract. Let m and n be fixed positive integers and M a right R-module. Recall that M is said to be (m,n)-injective if Ext 1 (P,M) = 0for any (m,n)-presented right R-module P; M is said to be (m,n)-flatif Tor 1 (N,P) = 0 for any (m,n)-presented left R-module P. In termsof some derived functors, relative injective or relative flat resolutions anddimensions are investigated. As applications, some new characterizationsof von Neumann regular rings and p.p. rings are given. 1. IntroductionLet C be a class of left R-modules and M a left R-module. Following ([7]),we say that a homomorphism ϕ : M → C is a C-preenvelope of M if C ∈ C andthe abelian group homomorphism Hom(ϕ,C ′ ) : Hom (C,C ′ ) → Hom (M,C ′ ) issurjective for each C ′ ∈ C. A C-preenvelope ϕ : M → C is called a C-envelopeif every endomorphism f : C → C such that fϕ = ϕ is an isomorphism. AC-envelope ϕ : M → C is said to have the unique mapping property (see[6]) if for any homomorphism f : M → C ′ with C ′ ∈ C, there is a uniquehomomorphism g : C → C