We investigate here the commutativity of a left (resp. right) s-unital ring R satisfying the polynomial identity yr [xny] xt = ±y3 [x, ym] (resp. yr [xn, y] xt = ± [x, ym] ys ) for some non-negative integers m >0, n > 0, r, s and t such that n + t > 1 (resp. m + s > 1 for r = 0). For such a ring R, we prove the commutativity if n + t > 1, and the commutators in R are n-torsion free (Q (n) property) for m > 1, n > 1, and ( t + 1)-torsion free for n = 1 (and t > 0). lf r = 0, then R is commutative provided m+ s > 1 and R has Q (m) property for m > 1, n > 1, and Q (s + 1) property for m = 1 (and s > 0). Especially, for r = 0, R is commutative, if m and n are relatively prime integers (not both equal to one).