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.