Commutativity Theorems for Rings With Constraints on Commutators

In the present paper we investigate commutativity of semiprime ring satisfying any one of the polynomial identities x^∫[xⁿ,y]y^r=±y^s[v^m,x] and x^∫[xⁿ,y]y^r=±[y^m,x]y^s for all x, y in R, where m, n, r, s and t are fixed non-negative integers, and further, we establish commutativity of rings with unity under some additional constraints. Moreover, it is also shown that the above result is true for s-unital ring. Finally, we provide some counter-examples which show that the hypotheses of our theorems are not altogether superfluous. The results of this paper generalize some of the well-known commutativity theorems for rings. (See [1], [2], [5], [9], [10], [12], [14].).