HYPERCENTRAL DERIVATIONS

Let k be a given nonnegative integer. Assume that R is a ring without nonzero nil two-sided ideals and that delta is a derivation of R with the property that, for any x is-an-element-of R, [delta(x(n(x)), x(n(x )]k = 0 for some integer n(x) greater-than-or-equal-to 1. Let U be the left Utumi quotient ring of R. It is proved here that there exists a central idempotent e of U such that, on the direct sum decomposition U = eU + (1 - e) U, the derivation delta vanishes identically on eU and the ring (1 - e) U is commutative. In particular, for any noncommutat ive prime ring R without nonzero nil two-sided ideals, a derivation de lta of R satisfying [delta(x(n(x)), x(n(x))]k = 0, n(x) greater-than-o r-equal-to 1, for all x is-an-element-of R, must vanish identically on R. (C) 1994 Academic Press, Inc.