Abstract
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 epsilon 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 epsilon 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.