In this paper, we generalize some well‐known commutativity theorems for associative rings as follows: Let n > 1, m, s, and t be fixed non‐negative integers such that s ≠ m − 1, or t ≠ n − 1, and let R be a ring with unity 1 satisfying the polynomial identity ys[xn, y] = [x, ym]xt for all y ∈ R. Suppose that (i) R has Q(n) (that is n[x, y] = 0 implies [x, y] = 0); (ii) the set of all nilpotent elements of R is central for t > 0, and (iii) the set of all zero‐divisors of R is also central for t > 0. Then R is commutative. If Q(n) is replaced by m and n are relatively prime positive integers, then R is commutative if extra constraint is given. Other related commutativity results are also obtained.