环的右奇异理想与交换性定理

Right singular ideals and commutativity theorems for rings

设R是一个有单位元的结合环.证明了如下结果:若对任意元x∈R\Zr(R)={x∈R|xZr(R)},y∈R,满足方程(xy)k=xkyk,k=n,n+1,n+2,其中n是一个正整数,则R是交换环.

Let R be an associative ring with identity. It is shown that if for each x∈R／Zr（R）={x∈R｜xZr（R）},y∈R and y∈R, （xy）^k=x^k y^k for k=n,n＋1,n＋2, where n is a positive integer, then R is commutative.