On Dual Automorphism-invariant Modules And ?-Projective Modules

This thesis mainly studies relative projectivity of modules from two directions.On the one hand,we generalize dual automorphism-invariant modules to dual auto-morphism invariant N-modules,and investigate their equivalent characterizations and properties under Morita duality.On the other hand,using the special submodule ?(N),we extend Rad-N-projective modules to ?-N-projective modules,and investigate some related rings.This thesis consists of three chapters.In the first chapter,we mainly introduce directions and trends on researches related to this thesis,and sum up the groundwork of this thesis.In the second chapter,we extend the dual automorphism-invariant modules to the dual automorphism N-invariant modules,obtain some properties of the dual automor-phism N-invariant modules,gives some equivalent characterizations of dual automor-phism N-invariant modules,which generalize related theorems about dual automor-phism invariant modules.Under Morita duality,for U-reflexive right R-modules M and N,we show that M(M*)is dual automorphism N-invariant if and only if M*(M)is automorphism N*-invariant.In particular,M(M*)is a dual automorphism-invariant module if and only if M*(M)is an automorphism-invariant module.In the third chapter,let ? be a class of right R-modules,using the particular sub-module ?(N)of N,we generalize Rad-N-projective modules to ?-N-projective mod-ules,give some properties and equivalent characterizations of ?-N-projective modules.We also consider closeness of direct sums of ?-N-projective modules and ?-projective relative to direct sums of modules.We introduce the concept of right ?-semisimple rings and give some equivalent characterizations of right ?-semisimple rings,as corol-laries,we obtain several usual properties of semisimple modules and rings.We also give the concept of right ?-hereditary rings and their equivalent characterizations.In particular,some equivalent properties of right hereditary rings are obtained as corol-laries.