Some Characterizations Of Abel Rings And Quasi-abel Rings

It is well-known that idempotents are very important in ring theory,and many famous rings are closely related to idempotents,such as strongly regular rings,exchange rings and clean rings.Since Abel rings and quasi-abel rings are defined by idempotents,the characterizations of them are necessary.In recent years,some conclusions about Abel rings have been reached,but many special properties remain to be fully explored.In this paper,the equivalence conditions for a ring R to be Abel or quasi-abel will be discussed through the properties of matrices and polynomials over ring R,and idempotents and generalized inverses of ring R.This paper is divided into three parts.In Chapter 1,we state the research background of Abel rings and quasi-abel rings.Also,some basic concepts and related conclusions are presented.In Chapter 2,we give some new characterizations of Abel rings.Firstly,by selecting and exploring some rings(sets)related to ring R,we describe a number of equivalent conditions for ring R to be Abel.For example,ring R is an Abel ring if and only if V2(R)is an Abel ring,V2(R)is a second order upper triangular matrix ring with the same diagonal elements over ring R.In addition,we discuss idempotents of ring R and give the proof of following conclusion:ring R is an Abel ring if and only if each idempotent of ring R can be uniquely represented as the sum of an involution and an idempotent.Finally,we use generalized inverses to characterize Abel rings.For instance,ring R is an Abel ring if and only if for any element a and idempotent e of ring R,when a?e exists,then a?ea=aa?e,a?e,is an inverse of a along e.In Chapter 3,we mainly focus on quasi-abel rings.First of all,we present some necessary and sufficient conditions for a ring R to be quasi-abel.For example,ring R is a quasi-abel ring if and only if UTM2(R)is a quasi-abel ring,UTM2(R)is a second order upper triangular matrix ring over ring R.Then we explore the properties of quasi-abel rings by discussing the relations among regular elements,group invertible elements and Moore-Penrose invertible elements of ring R.For example,suppose ring R is a quasi-abel ring,then each regular element of ring R is a group invertible element.