On (m,n)-Injective Modules and (m,n)-Coherent Rings

Abstract

Let R be a ring. For two fixed positive integers m and n, a right R-module M is called (m,n)-injective in case every right R-homomorphism from an n-generated submodule of R m to M extends to one from R m to M. R is said to be left (m,n)-coherent if each n-generated submodule of the left R-module R m is finitely presented. In this paper, we give some new characterizations of (m,n)-injective modules. We also derive various equivalent conditions for a ring to be left (m,n)-coherent. Some known results on coherent rings are obtained as corollaries.