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].