Abstract :
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 set membership, variant R, [δ(xn(x)), xn(x)]k = 0 for some integer n(x) ≥ 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 = eUcircled plus(1 − e) U, the derivation δ vanishes identically on eU and the ring (1 − e) U is commutative. In particular, for any noncommutative prime ring R without nonzero nil two-sided ideals, a derivation δ of R satisfying [δ(x(n(x)), x(n(x))]k = 0, n(x) ≥ 1, for all x set membership, variant R, must vanish identically on R.
