About M-Principally Injective Modules, Quasi-Principally Injective Modules, Pseudo-Principally Injective Modules

This thesis is made up of three chapters:In the first chapter,some equivalent conditions of M-principally injective modules are given by using the concepts of M-principally injective modules:N is M-principally injective if and only if kers(?)kerφimpliesφ∈Ts for all s∈S=End(M_R)andφ∈T=Hom_R(M,N),and etc.And it is proved thatΠ_AN_αis M-principally injective module if and only if(?)_α∈A,N_αis M-principally injective modules by its equivalent conditions;Moreover,some characterizations of M-principally injective modules are given,which are generalized some results of quasi-principally injective modules.In the second chapter,it is defined pseudo-principally injective modules, and some equivalent conditions of pseudo-principally injective modules are given;Moreover,some characterizations of pseudo-principally injective modules are given,and it is proved that pseudo-principally injective modules satisfy C₂ and C₃ conditions,and etc;Next,the endomorphism ring of pseudo-principally injective modules and its Jacobson Radical are considered,and some characterizations on its Jacobson Radical are given, and it is proved that the endomorphism ring of pseudo-principally injective modules is local ring under special conditions;At last,it is proved by the definitions of M-annihilator submodules and t-nilpotently on M that the Jacobson Radical of the endomorphism ring of pseudo-principally injective modules is nilpotent under special conditions,In the third chapter,some equivalent conditions of quasi-principally injective modules are given by using the concept of quasi-principally injective modules;Next,the relationship between quasi-principally injective modules and pseudo-principally injective modules is discussed,and it is proved that pseudo-principally injective modules and quasi-principally injective modules are equivalent under special conditions.