The main result of the paper is the following theorem:

Theorem. Let R be a ring. Assume that for any two elements x and y in R, there exists an integral polynomial f(t)=t-t2g_(x,y)(t) such that

                   f(xy)=f(yx),

then R is commutative.

Corollary. If for any x and y in R, there exists an integer n(x,y)>1 such that

        xy-yx=(xy)^n(x,y)-(yx)^n(x,y),

then R is commutative. 

This result generalized a theorem of Niu Fengwen and Guo Yuanchun (Jining University) and a well-known result of N. Jacobson.

Note that I. N. Herstein obtained a well-known commutativity condition for any R, his condition is

              xy-yx=(xy-yx)^n(x,y),

where n(x,y)>1.