半质环的两个交换性定理

Two commutative theorem of semi-prime rings

证明了满足下列条件的半质环是交换环:1)若对x,y,z∈R,存在整数m=m(x,z)>1,n=n(x,z)>1,使得[(xmy)n-xym,z]∈Z(R)则R为交换环.2)若对x,y,z∈R,存在整数m=m(y,z)>1,n=n(y,z)>1,使得[(xmy)n+xmy,z]∈Z(R)则R为交换环.

In this paper,two commutativity theorem on semiprimerings as follows： 1 ） For any x,y,z∈ R are proved,there exist integers m = m（x,z）〉1,n=n（x,z）〉1m such that [ （x^m y）^n -xy^m ,z] ∈ Z（R） then R is commutative. 2） For any x,y,z ∈ R, there exist integers m = m （y,z） 〉 1, n = n （y,z） 〉 1, such that [ （x^m y）^n ＋x^m y,z] ∈Z（R） then R is commutative.