Some Properties Of R{D,C} Ring

In recent years,R[D.C]ring has become an important research object in algebra.Con-structing other rings and some important counterexamples or giving the equivalent descriptions of some rings by R[D,.C]ring have become an important method in ring theory.On the basis of these study,we extend the research content.These are some main research directions in this paper:Firstly,The conditions of R{D,C},R(D,C)and R[D,C]to become quasi-morphic ring,weakly quasi-morphic ring,2-primal ring,zip and p.p.-ring are studied;Secondly,we discuss the nature of expansion ring R{D,C},including R{D,C} to be a-Armendariz ring,skew ?-meCoy ring,feckly almost strongly II-regular ring,and the research of the expand model ring R[D,C].This thesis mainly consists of the following components:Chapter 1,In this chapter,we introduce the historical background,development process and research status of R{D,C} ring,and briefly summarize the main work and important results of this paper.Chapter 2,The definitions and concepts used in this paper are given.Chapter 3,In this chapter,some properties of ring R{D,C} are studied.It is proved that:(1)R{D,C} is a weakly quasi-morphic ring if and only if D is a weakly quasi-morphic ring,and for any x E C there exists y E C such that Cx = Lc(y),Dx = lD(y);(2)if R{D,C} is a 2-primal ring,then D and C are both 2-primal ring;(3)S = R[D,C]is a right zip ring if and only if D and C are both right zip ring.Chapter 4,We discuss the nature of the expansion ring R{D,C}.It is proved that:(1)S = R{D,C} is an ?-Armendariz ring if and only if D is an ?-Armendatriz ring;(2)if R{D,C} is a VNL-ring,then D is a regular ring and C is a VNL-ring;(3)S = R[D,C]is a feckly almost strongly ?-regular ring if and only if D is a feckly strongly ?-regular ring and C/J(D)?J(C)is an almost strongly ?-regular ring.Chapter 5,We give a summary about the property of ring R{D,C}?R(D,C)?R[D,C]and its equivalent descriptions,and we have made a further outlook about the research direction.