On Lie ideals and Jordan left derivations of prime rings

Let R be a 2-torsion free prime ring and let U be a Lie ideal of R such that u2 ∈ U for all u ∈ U . In the present paper it is shown that if d is an additive mappings of R into itself satisfying d(u 2) = 2ud(u) for all u ∈ U , then d(uv) = ud(v) + vd(u) for all u, v ∈ U .