半质环的一个交换性定理

A commutativity theorem of semiprime rings

设R为有单位元1的半质环,C为R的中心,k为大于1的整数。若对任意非幂零元x∈R及任意y∈R有(xy)~k-x^ky^k∈C或(xy)~k-y^kx^k∈C,则R为交换环。

Let R be a semiprime ring with unity 1 and C be the center of R, k be an integer greater than 1. If for any non-nilpotent element x R and any y R R, there exist (xy)k-xkyk C or (xy)k-ykxk C,then R is a commutative ring.