The main theorem of this paper is that a ring R with unity is commutative if and only if there is a nil subset B of R such thatl. for each x ∊ R, either x ∊ Z(R) or there is a polynormial f over Z with x - x2 f (x)  ∊ B;2. for each x, y x ∊ R, there are non-negative integers n > 1, m, r, s depending on a pair of ring elements x,y with x(xmy ± xrynxs) - (xm y ± xrynxs)x = 0.A related result for a nil commutative subset of R is given and the restrictions on the hypothesis of our result are justified by examples.