characterization of (δ, ε)-double derivations on rings and algebras

This paper is an attempt to prove the following result:let n>1 be an integer and let r be a n!-torsion-free ring with the identity element. suppose that d,δ,ε are additive mappings satisfying d(xn)=∑j=1nxn−jd(x)xj−1+∑j=1n−1∑i=1jxn−1−j(δ(x)xj−iε(x)+ε(x)xj−iδ(x))xi−1 for all x∈r. if δ(e)=ε(e)=0, then d is a jordan (δ,ε)-double derivation. in particular, if r is a semiprime algebra and further, δ(x)ε(x)+ε(x)δ(x)=12[(δε+εδ)(x2)−(δε(x)+εδ(x))x−x(δε(x)+εδ(x))] holds for all x∈r, then d−δε+εδ2 is a derivation on r.