On certain subrings and ideals of prime rings

Let R be a noncommutative prime ring of characteristic not 2 and let D be a nonzero derivation of R. A theorem of Breshar and Vukman asserts that U, the subring of R generated by all [r(D), r], r is an element of R, contains a nonzero left ideal of R and a nonzero right ideal of R. We prove a more general result stating that U contains a nonzero two-sided ideal of R thus answering in a positive way on a question of Breshar and Vukman.