SOME COMMUTATIVITY THEOREMS FOR RINGS

In this paper we prove two results: (1) If, in a semiprime ring R, for x, y is-an-element-of R, there exist fixed positive integers n, m greater than 1 such that either (a) [x(n), y(m)] is-an-element-of Z(R) or (b): (x(n) o y(m)) is-an-element-of Z(R) then R is commutative. (2) If, in a division ring, R, for x, y is-an-element-of R. there exists a positive integer n > 1 such that (xy)n y(n) - y(n)(yx)n commutes with y, then R is commutative.