A note on (σ,τ)-derivations in prime rings

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