Pseudo Principally Quasi-Injective Modules And Pseudo Quasi-Principally Injective Modules

Abstract

The purposes of this thesis are to (1) study the properties and characterizations of pseudo principally quasi-injective modules and pseudo quasi-principally injective modules, (2) study the properties and characterizations of endomorphism rings of the two types of modules, (3) extend the concepts of principally quasi-injective modules and quasi-principally injective modules and (4) find some relations between among the four types of modules mentioned. 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 principally M-injective if every R-homomorphism from a principal submodule of M to N can be extended to an R-homomorphism from M to N. A right R-module M is called principally quasi-injective if it is principally M-injective. 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 principally quasi-injective modules and quasi-principally injective modules are extended to be pseudo principally quasi-injective modules and pseudo quasi-principally injective modules, respectively. A right R-module N is called pseudo principally M-injective if every R-monomorphism from a principal submodule of M to N can be extended to an R-homomorphism from M to N. A right R-module M is called pseudo principally qausi-injective if it is pseudo principally M-injective. A right R-module N is called pseudo M-principally injective if every R-monomorphism 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 pseudo qausi-principally injective if it is pseudo M-principally injective…