Strong 2-Sum Property Of Matrix Rings And ~*-cleanness Of Finite Group Rings

In recent decades, it is one of active topics in ring theory to study properties and structure of rings by using properties of special elements of rings.The first chapter introduces the research background of this thesis and related to some basic concepts and preliminaries, finally summarizes main results of this thesis.The second chapter is mainly to study the strong 2-sum property of local rings and its extensions. The notion of a strong 2-sum ring was introduced by Tang and Zhou in 2013. We first summarize the background and development of the study of strong 2-sum rings in Section 2.1. In Section 2.2, we enumerate some basic properties of strong 2-sum rings. We consider the strong 2-sum property of local rings and its extensions, including power series extension, trivial extension and matrix extension in Section 2.3.In particular, we prove for a commutative local R, the strong 2-sum property of R[[x]],(7)(8)2(44)R,(7)(8)3(44)R,(7)(8)nT R just depend on the strong2-sum property of R. In Section 2.4, we give two strong 2-sum subrings of full matrix ring of order 3. In Section 2.5, we sum up investigations on the strong 2-sum property of rings, and give some conjectures and problems.The third chapter is mainly to study *-cleanness of finite group rings.The concepts of *-clean rings was introduced by Vas in 2010. We first summarize status of the study of *-clean rings in Section 3.1. In Section3.2, we prove that ifqF is a finite field of order q, G is a finite abelian group of order n and gcd(q,n)(28)1, thenqF G is *-clean if and only if there exists a positive integer ? such that q1(mod m)??-,where m(28)exp(G), the exponent of G. In particular, if G is a finite cyclic group, then an open question about *-cleanness of finite group rings will be settled fully. Finally, we extend the result to any finite abelian group G. In Section 3.3, we sum up main results of third chapter and put forward some open problems at same time.