Commutativity of rings with polynomial constraints

Let p, q and r be fixed non-negative integers. In this note, it is shown that if R is left (right) s-unital ring satisfying [f(x(p)y(q))-x(r)y,x]=0 ([f(x(p)y(q))-yx(r), X]=0, respectively) where f (lambda) is an element of lambda(2) Z[lambda], then R is commutative. Moreover, commutativity of R is also obtained under different sets of constraints on integral exponents. Also, we provide some counterexamples which show that the hypotheses are not altogether superfluous. Thus, many well-known commutativity theorems become corollaries of our results.