Commutative Results for Rings

Abstract

In this paper, we provided two commutativity theorems are : If R is a semi prime ring and there exist a fixed positive integer $m>1$ such that either (i) $[[a, b]^{m}-[a^{m}, b^{m}], a]=0$ or $[(a \circ b)^{m}-(a^{m} \circ b^{m}), a] = 0$, then R is commutative ring. (ii) For all a, b in R there exists a positive integer $m = m(a, b) >1$ such that $(a b)^{m}= b a$, then R is commutative ring.