满足方程xy-yx-(xy-yx)^nZ的环的结构

Structure of rings satisfying equation xy-yx= （xy-yx）^nz

借助于某种换位子等式,给出SZC环的定义,研究SZC环的一些性质.主要证明了如下结果:①SZC环是CN环和ZC环;②R为强正则环当且仅当R为SZC环和正则环;③设R为SZC环且C(R)≠R,若R为素环,则R为交换环;④R为Abel环当且仅当对任意e∈E(R),任意x∈R,存在n=n(e,x)>1,z=ze,x∈R,使得ex-xe=(ex-xe)nz;⑤R为CN环当且仅当对任意x∈N(R),任意y∈R,存在n=n(x,y)>1,z=zx,y∈N(R),使得xy-yx=(xy-yx)nz.

Using certain identities on commuattors of rings, the author gives the definition of SZC rings and introduces some properties of SZC rings. The following results are obtained： ① SZC rings are CN and ZC; ②R is a strongly regular ring if and only if R is a regular ring and SZC ring; ③Let R be a SZC ring and C（R）≠R. If R is prime, then R is commutative; ④ R is an Abel ring if and only if for any eEE（R）, z∈R, there exist n-n（e,x）〉1,z=zx.y,∈R, such that ez-zg-（ex-xe）^nz; ⑤ R is a CN ring if and only if for any x∈N（R）, y∈R, there exist n-n（x,y）〉1,z=zx.y∈N（R）, such thatzy-yz-（xy-yx）^nz.