[en] We study the commutativity of certain class of rings, namely rings with unity 1 and right s-unital rings under each of the following properties [yxm - xn f(y)xp,x]=0, [yxm+xnf(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. (author). 14 refs