Generalized Derivations of Rings and Banach Algebras

Let R be a prime ring with Utumi quotient ring U and m, n fixed positive integers. If R admits a generalized derivation F associated with a derivation d such that (F([x, y]))m = [x, y]n for all x, y ∈ R, then R is commutative or d = 0. Moreover, we also examine the cases R is a semiprime ring. Finally, we apply the above result to Banach algebras, and we obtain a noncommutative version of the Singer–Wermer theorem.