关于环的交换性条件

On Commutativity Conditions for Rings

设R是有单位元的结合环,I,J分别是R的补右零化子集和补左零化子集,Z_(r)(R),Z_(l)(R)分别是R的右奇异理想和左奇异理想.证明了如果存在互素的正整数m,n,使得任意x∈RI,y∈RJ均满足(xy)^(k)=x^(k)y^(k),其中k=m,m+1,n,n+1;或者任意x∈RJ,y∈RI均满足(xy)^(k)=y^(k)x^(k),其中k=m-1,m,n-1,n是正整数,那么R是交换环.特别地,如果对于I=N(R)∪J(R)∪Z_(r)(R)和J=N(R)∪J(R)∪Z_(l)(R),存在正整数m,使得任意x∈RI,y∈RJ均满足(xy)^(k)=x^(k)y^(k),其中k=m,m+1,m+2;或者任意x∈RJ,y∈RI均满足(xy)^(k)=y^(k)x^(k),其中k=m-1,m,m+1,那么R是交换环.

Let R be an associative ring with identity,I and J be a complementary right annihilator and a complementary left annihilator of R,respectively,and let Z_(r)(R)and Z_(l)(R)be the right singular ideal and the left singular ideal of R,respectively.It is proved that if there exist coprime positive integers m and n such that(xy)^(k)=x^(k)y^(k)for each x∈R\I and each y∈R\J,where k=m,m+1,n,n+1,or(xy)^(k)=y^(k)x^(k)for each x∈R\J and each y∈R\I,where k=m-1,m,n-1,n are positive integers,then R is a commutative ring.Especially,if I=N(R)U J(R)U Z_r(R)and J=N(R)U J(R)U Z_(l)(R),and there exists a positive integer m such that(xy)^(k)=x^(k)y^(k)for each x∈R\I and each y∈R\J,where k=m,m+1,m+2 or(xy)^(k)-y^(k)x^(k)for each x∈R\J and each y∈R\I,where k=m-1,m,m+1,then R is a commutative ring.