Jordan k-Derivations on Lie Ideals of Prime Γ-Rings

Let M be a Γ- ring and U a Lie ideal of M. Let d : M → M and k :Γ → Γ be additive mappings. Then d is a k- derivation on U of M if d(uαv) = d(u)αv + uk(α)v + uαd(v) is satisfied for all u, v ∈ U and α ∈ Γ. And d is a Jordan k- derivation on U of M if d(uαu) = d(u)αu + uk(α)u + uαd(u) holds for all u ∈ U and α ∈ Γ. It is well-known that every k- derivation on U of M is a Jordan k- derivation on U of M but the converse is not true in general. In this article we prove that every Jordan k- derivation on U of M is a k- derivation on U of M if , M is a 2- torsion free prime Γ- ring and U is a Lie ideal of M such that uαu ∈ U for all u ∈ U and α ∈ Γ.