ON DERIVATIONS AND COMMUTATIVITY IN SEMIPRIME RINGS

Let R be a ring, Z its center, TJ 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)]] is an element of Z for all x is an element of U; (ii) for a fixed positive integer n, [x(n), D(x)] is an element of Z for all x is an element of U.