Further Studies Of The Quasi-morphic Property Of Groups And Modules

A modules M is called quasi-morphic,if for each α∈End(M),there exists β∈End(M) and γ∈End(M) such that Mα=Kerβ and Kera=Mγ. Further more,the quasi-morphic property can also be extended to groups easily if groups are consid-ered to be Z-modules.In this dissertation,we systematically study the properties of quasi-morphic groups and quasi-morphic modules.In chapter one,the background and current research status of mor-phic groups,morphic rings and morphic modules are reviewed.In chapter two,the quasi-morphic groups are investigated.First-ly.some characterizations of quasi-morphic group are obtained from a different view of paper [9]. Secondly,the notion and some properties of strong quasi-morphic group are given. And we point out the finitely generated Abelian group is strong quasi-morphic if and only if it is a finite group. In the end of this part,the quasi-morphic Abelian groups are investigated. We show that the quasi-morphic divisible group is equivalent to the torsion-free group and get the characterizations of twisted group’s quasi-morphic property.In chapter three,the quasi-morphic modules are studied. First-ly,the quasi-morphic endomorphism is investigated. Some examples and equivalent conditions of the quasi-morphic endomorphism are giv-en. Secondly.some characterizations and properties of quasi-morphic module are obtained. It is proved that the sets of endomorphic images of quasi-morphic modules are moduler. Thirdly,the notion and some characterizations and their properties of P-Quasi-morphic module are given. Lastly,based on the paper [8],endomorphism rings of the quasi-morphic modules are studied and one characterization of the regular ring is given.