Commutativity for a Certain Class of Rings

We first establish the commutativity for the semiprime ring satisfying x ⁿ , y]x ^r = ±y ^sx, y ^m]y ^t for all x, y in R, where m, n, r, s, and t are fixed non-negative integers, and further, we investigate the commutativity of rings with unity under some additional hypotheses. Moreover, it is also shown that the above result is true for s-unital rings. Also, we provide some counterexamples 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 which are right s-unital.