MININJECTIVITY AND KASCH MODULES

Abstract

Let R be an associate ring with identity. A right R-module M is called mininjective if every homomorphism from a simple right ideal of R to M can be extended to R. We now extend this notion to modules. We call a module N an M-mininjective module if every homomorphism from a simple M-cyclic submodule of M to N can be extended to M. In this note, we characterize quasi-mininjective modules and show that for a finitely generated quasi-minjective module M which is a Kasch module, there is a bijection between the class of all maximal submodules of M and the class of all minimal left ideals of its endomorphism ring S = End(M) if and only if SrM(K)=K for any simple left ideal K of S. The results obtained by Nihcolson and Yousif in mininjective rings are generalized.