WCN Ring

Researching the conditions for a von Neumann regular ring to be a strongly regular ring is an important part of algebraic ring-theoretic research. Hence lots of rings are introduced, for example, reduced rings, reversible rings, semicommutative rings, Abel rings and so on.In this paper, a new class of rings, so-called WCN ring, is introduced, which is between semicommutative rings and CN rings, many properties of semicommutative rings and CN rings are inherited. According to the research on WCN rings, we not only give some new characteriza-tions of strongly regular rings, but also aim at finding out the relationship among the rings mentioned above. Meanwhile, some applications to WCN rings are given, which generalize some known results.In the whole, this paper has four chapters. The first chapter introduces the study background of WCN rings and some preliminaries needed in the paper.In the second chapter, we mainly study some examples, and point out the relationship among WCN rings, CN rings, reduced rings, NCI rings and semicommutative rings. At the same time, we enumerate and prove some basic properties of WCN rings. We’ve got the conclusions as follows:(1) R is a reduced ring if and only if R is a CN ring and T2(R) is a WCN ring.(2) R is a WCN ring if and only if for any a∈N(R), b∈R, ab=0implies aRb=0, or there exists c∈R such that0≠acb-(acb)n∈Z(R) for any positive integer n≥2.(3) R is a CN ring if and only if for any a∈N(R), there exists a positive integer n≥2such that a-an∈Z(R).(4)R is a reduced ring if and only if R is a left NPP ring、WCN ring and left idempotent reflexive ring if and only if R is a left NPP CN ring.In the third chapter, we study the regularity of WCN rings. We mainly prove some results as follows:(5) If R is a von Neumann regular WCN ring, then R is a strongly regular ring.(6) Left SF WCN rings are strongly regular rings.(7) Let R be a left MC2WCN ring, if every simple singular left R-module is nil-njective, then R is a reduced ring.In the fourth chapter, we mainly study the Exchange properties of WCN rings. We’ve got the results as follows:(8) If R is a WCN ring, then (a) R is a weakly Exchange ring if and only if R is a weakly Clean ring;(b) R is a Exchange ring if and only if R is a Clean ring.(9) WCN Exchange rings have stable range1.