On the structure of prime and semiprime rings with generalized skew derivations

Let Formula: see text be a ring of characteristic different from Formula: see text, Formula: see text fixed positive integers, Formula: see text a noncentral Lie ideal of Formula: see text and Formula: see text a nonzero generalized skew derivation of Formula: see text. We prove the following results: (a) If Formula: see text is prime and there exists Formula: see text such that Formula: see text then Formula: see text, the Formula: see text matrix ring over a field Formula: see text. (b) If Formula: see text is semiprime and Formula: see text then either Formula: see text centralizes a nonzero ideal of Formula: see text or Formula: see text is a polynomial identity for Formula: see text.