Small Simple Quasi-injective Modules

Abstract

The purposes of this thesis are to (1) study properties and characterizations of small simple quasi-injective modules, (2) study properties and characterizations of endomorphism rings of small simple quasi-injective modules, (3) extend the concept of small principally quasi-injective modules, and (4) find some relations between small simple quasi-injective modules, small principally quasi-injective modules and projective modules. Let R be a ring. A right R-module M is called mininjective if, for each simple right ideal K of R, every R-homomorphism γ: K → M extends to an R-homomorphism from R to M. A right R-module N is called small principally M-injective if every R-homomorphism from a small and principal submodule of M to N can be extended to an R-homomorphism from M to N. A right R-module M is called small principally quasi-injective if it is small principally M-injective. The notion of small principally quasi-injective modules is extended to be small simple quasi-injective modules. A right R-module N is called small simple M-injective if every R-homomorphism from a small and simple submodule of M to N can be extended to an R-homomorphism from M to N. A right R-module M is called small simple quasi-injective if it is small simple M-injective.