On certain annihilators in prime and semiprime rings with derivations

Abstract

Let R be a prime ring, a ∈ R, I = 0 an ideal ofR and d : R −→ R be a derivation of R. We prove that: i) if for any x,y ∈ I, a(d(xy)−xy) = 0, then a = 0; ii) if a(d(xy) − xy) ∈ Z(R), for any x,y ∈ I, thenR is commutative. We also examine the case when R is a semiprime ring.