Derivations and Right centralizers on 2-torsion free prime rings

Abstract

Let R be a 2-torsion free prime ring, d be a derivation of R and T be a non zero right centralizer of R. In the present paper, we investigate the commutativity of R admitting derivation d and right centralizer T satisfying any of the following properties, for all x, y ∈R:
(i) T([x, y]) ± [T(x), T(y)]) = 0,
(ii) T([x, y]) ± [d(x), d(y)]) = 0,
(iii)T([x, y]) + [d(x), T(y)]) = 0,
(iv) d ([x, y]) ± T([x, y]) = 0,
(v) d ([x, y]) + [d(x), T(y)] = 0.