On derivations and commutativity in semiprime rings

Abstract

Let R be a ring, Z its center, U a nonzero left ideal, and D:R → R a derivation. We show that if R is semiprime with suitably-restricted additive torsion, then R must contain nonzero central ideals if one of the following holds: (i) [x, [x, D(x)]] ∊ Z for all x ∊ U; (ii) for a fixed positive integer n, [xn, D(x)] ∊ Z for all x ∊ U