A Note on An Engel Condition with Generalized Derivations in Rings

Let R be a prime ring with characteristic different from two, I be a nonzero ideal of R, and F be a generalized derivation associated with a nonzero derivation d of R. In the present paper we investigate the commutativity of R satisfying the relation F([x, y]k) n = ([x, y]k) l for all x, y ∈ I, where l, n, k are fixed positive integers. Moreover, let R be a semiprime ring, A = O(R) be an orthogonal completion of R, and B = B(C) be the Boolean ring of C. Suppose F([x, y]k) n = ([x, y]k) l for all x, y ∈ R, then there exists a central idempotent element e of B such that d vanishes identically on eA and the ring (1 − e)A is commutative.