Abstract
Let R be a prime ring and set [x, y](1) = [x, y] = xy - yx for all x, y is an element of R and inductively [x, y](k) = [[x, y](k-1) , y] for k > 1. We apply the theory of generalized polynomial identities with automorphism and skew derivations to obtain the following result: Let R be a prime ring and I a nonzero ideal of R. Suppose that (delta, phi) is a skew derivation of R such that delta([x, y]) = [x, y](n) for all x, y is an element of I, then R is commutative.