We study the commutativity of certain class of rings, namely rings with unity 1 and right s-unital under each of the following properties [yxm - xn f (y) xP , x]= 0, [yxm + xn f (y) xP, x] = 0, where f (t) is a polynomial in t2Z [t] varying with pair of ring elements x, y and m, n, p are fixed non-negative integers. Moreover, the results have been extended to the case when m and n depend on the choice of x and y and the ring satisfies the Chacron's Theorem.