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 Ext 1 R (R/I, M) = 0 for any maximal left ideal I. A left R-module M is called max-projective (or m-projective for short) if Ext 1 R (M, N) = 0 for any max-injective left R-module N. We prove that every left R-module has a special m-projective precover and a special m-injective preenvelope. We characterize C-rings, SF rings and max-hereditary rings using m-projective and m-injective modules.

On MF-projective modules

“…, where k is a field, and X denotes the residue class of X in R. Then every right R-module is (S)M F -projective by Proposition 2.11, so is the ideal X, in particular. However X is not projective, because X 2 = 0 implies that X is not a free ideal in the local ring R. Recall that a ring R is called left max-hereditary if every maximal left ideal is projective (see [1]). This is equivalent to saying that every factor of a max-injective left R-module is max-injective (see [ (1) A is projective.…”

“…Otherwise, since R is left Noetherian, every simple left R-module is finitely presented. If R was a left SF-ring, then every simple left R-module would be projective by[22, Corollary 3.58], whence R would be semisimple, a contradiction.We shall now give a condition for the converse of Remark 2.2(1). Let R be a left max-hereditary ring or a left SF -ring.…”