T-nilpotent Rings,Directly Finite Rings And J-nilpotent Rings

T-nilpotency is an important property of a ring.Some basic properties of a perfect ring depend on the generalization of the concept of T-nilpotency.Directly finite rings play an important role in the study of partially ordered Abelian groups and von Neumann regular rings.The Jacobson radical theory of rings is a good extension of Artinian ring theory,which plays an important role in studying the structure of rings.In this paper,we mainly investigate the extensions of T-nilpotent rings and directly finite rings.Furthermore,we also study some properties of rings satisfying Jacobson radical contained in the set of all nilpotent elements.This thesis mainly consists of the following components:Chapter 1:We introduce the background,development process and research status of T-nilpotent ring,directly finite ring and Jacobson radical theory,and briefly summarize some important work and results in the literature.Chapter 2:We introduce some essential concepts,such as nil rings,nilpotent rings,sym-metric rings,reversible rings,semicommutative rings,J-reduced rings,JU rings,UU rings,weakly JU rings,weakly UU rings,nil-clean rings and other rings,and some results which were frequently used in the sequel.Chapter 3:We mainly investigate some extension properties of T-nilpotent rings.The main results are the following:If R is a ring and a an automorphism of R,then R is a left T-nilpotent ring if and only if the skew polynomial ring R[x;?]over R is a left T-nilpotent ring,if and only if the Nagata extension over R is a left T-nilpotent ring,if and only if the skew triangular matrix ring Tn(R,?)over R is a left T-nilpotent ring.Chapter 4:We mainly investigate some extension problems of directly rings.The main results are the following:Under certain conditions,R is a directly finite ring if and only if the skew power series ring R[[x;?]]over R is a directly finite ring,if and only if the trivial extension T(R,M)over R is a directly finite ring,if and only if the skew triangular matrix ring Tn(R,?)over R is a directly finite ring.Chapter 5:We intruduce a new concept of J-nilpotent rings.A ring R is called J-nilpotent if Jacobson radical contained in the set of all nilpotent elements.we discuss the relationships between J-nilpotent rings and related rings,and consider some extensions of J-nilpotent rings.Chapter 6:We give a summary about the properties of these classes of rings in this article,and we have made a further outlook about the future research directions.