Some Studies Of GWCN Rings

In 1986, Johnsen, Qutcalt and Yaqup in the literature [1] proved the following:Let R be a ring. If (xy)2=xy2x for any x,y∈R, then R is a commutative ring.Inspired by this theorem, we give the definition of GWCN rings, which are between CN rings and nil-semicommutative rings. As is known to all, researching the conditions for a von Neumann regular ring to be a strongly regular ring is an important part of algebraic ring —theoretic research. According to the research on GWCN rings, we not only give some new characterizations of GWCN rings and point out the relationship among reduced rings, NI rings, SF rings and Abel rings, but also give some characterizations of left min-abel rings and strongly regular rings. Meanwhile, we discuss some signficant properties of exchange GWCN rings.In the whole, this paper has five chapters. The first chapter introduces the background of GWCN rings and some preliminaries needed in the paper.In the second chapter, we mainly study some examples, and point out the relationship among GWCN rings, CN rings, reduced rings, nil-semicommutative rings and Abel rings. At the same time, we enumerate and prove some basic properties of GWCN rings. We’ve got the conclusions as follows:(1)R is a reduced ring if and only if T2(R) is a GWCN ring if and only if Z3(R) is a GWCN ring.(2)If I is a reduced ideal of R and R/I is a GWCN ring, then R is GWCN.(3)Let R be a GWCN ring and idempotents can be lifted modulo J(R), then R/J(R) is Abelian.In the third chapter, we mainly study the regularity of GWCN ring. It is known that the reduced von Neumann regular ring is a strongly regular ring. We generalize this result as follows:If R is a von Neumann regular GWCN ring, then R is a strongly regular ring.Reseaching the conditions for SF ring to be a von Neumann ring has always been a hot spot of ring-theoretic research. The famous result given by professor Rege in [2] is that the reduced SF ring is a strongly regular ring. We prove that R is a strongly regular ring if and only if V2(R) is a GWCN ring and R is a left SF ring. Meanwhile, with the help of GWCN rings, we give some characterizations of reduced rings. We prove that R is a reduced ring if and only if R is a left NSF ring and R[x]/(x2) is a GWCN ring if and only if R is a left NSF ring and R∝ R is a GWCN ring.In the fourth chapter, through the study of GWCN rings, we give some new properties of left min-abel rings. We prove thatR is a left min-abel ring if and only if for any k ∈M, (R)∩N(R) and x∈R,we haveMeanwhile, we also prove that if R is a left MC2 GWCN ring and every simple single left R-module is YJ-injective, then R is a reduced weak regular ring. So we generalize the relust belonging to Kim, Nam and Kim [3] that if R is a ZI ring whose every simple single left module is YJ-injective, then R is a reduced weak regular ring.In the last chapter, we mainly study the exchange properties of GWCN rings. We’ve got the results as follows:(1) If R is a GWCN exchange ring and p is a prime ideal of R, then R/P is a local ring. (2) If R is a GWCN ring, then R is a π-regular ring if and only if R is an NI ring and R/N(R) is regular. It is known that clean ring must be exchange, but the coverse is not true. In 1977, Nicholson [4] pointed out that Abelian exchange ring is clean. In 1995, Yu Hua ping [5] pointed out the left quasi-duo exchange ring is clean. In this chapter, we proved that if R is a GWCN exchange ring, then R is quasi-duo and clean.