Quasi-Small Principally-Injective Modules

Abstract

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