JORDAN HIGHER DERIVATIONS, A NEW APPROACH

Let A be a unital algebra over a 2-torsion free commutative ring R and M be a unital A-bimodule. We show taht every Jordan higher derivation D = {Dn}nN0 from the trivial extension A⋉M into itself is a higher derivation, if PD1(QXP)Q = QD1(PXQ)P = 0 for all X ∈ A⋉M, in which P = (e,0) and Q = (e′,0) for some non-trivial idempotent elements e ∈ A and e′ = 1−e satisfying the following conditions: eAe′Ae = {0}, e′AeAe′ = {0}, e(l.annM)e = {0}, e′(r.annM)e′ = {0} and eme′ = m for all m ∈ M.