A note on (?, ?)-derivations in prime rings

Let R be a 2-torsion free prime ring and let ?, ? be automorphisms of R. For any x, y ? R, set [x, y]?, ? = x?(y) - ?(y)x. Suppose that d is a (?, ?)-derivation defined on R. In the present paper it is shown that (i) if d is a nonzero (?, ?)-derivation and h is a nonzero derivation of R such that dh(R) ? C?, ? then R is commutative, (ii) if R satisfies [d(x), x]?, ? ? C?, ?, then either d = 0 or R is commutative. (iii) if I is a nonzero ideal of R such that d(xy) = d(y/x) for all x, y ? I, then R is commutative.