关于环的交换性定理

On Commutativity Theorems for Rings

令R为有单位元的结合环,M(R)=N(R)∪J/(R).证明了如果存在正整数m使得所有x,y∈_R\M(R)均满足(xy)~k=x^ky^k(其中k=m,m+1,m+2);或者使得所有x,y∈R\M(R)均满足(xy)~k=y^kx^k(其中k=m-1,m,m+1为正整数),那么R是交换环.

Let R be an associative ring with identity,M(R)=N(R)∪J(R).It is proved that if there exists a positive integer m such that(xy)~k=x^ky^k for all x,y∈R\M(R)(where k=m,m+1,m+2);or such that(xy)~k=y^kx^k for all x,y∈R\M(R)(where k=m-1,m,m+1 are positive integers),then R is a commutative ring.