In this paper, we generalize sone well-known commutativity theorems for associative rings as follows: Let ’, > 1. ,,, .,, and be fixed nou-ncgative integers such that s ik m1, or i/= n1, and let R be a ring xvith unity satisfying the polynomial identity y*[x’,y] [x,y’]x for all y R. Sul,lose that (i) R has Q(z) (that is n[x,y] 0 implies [z,y] 0); (ii) the set of d] nilpotent ,,lem,’nts of R is ...