On M-injective and M-projective Modules

Abstract

A left R-module M is called max-injective (or m-injective for short) if for any maximal left ideal I,any homomorphism f : I → M can be extended to g : R → M, if and only if Ext1R(R/I, M) = 0for any maximal left ideal I. A left R-module M is called max-projective (or m-projective for short)if Ext1R(M, N) = 0 for any max-injective left R-module N. We prove that every left R-module has aspecial m-projective precover and a special m-injective preenvelope. We characterize C-rings, SF ringsand max-hereditary rings using m-projective and m-injective modules.