JORDAN HIGHER DERIVATIONS, A NEW APPROACH

‎Let \mathcal{A} be a unital algebra over a ۲-torsion free commutative ring \mathcal{R} and \mathcal{M} be a unital \mathcal{A} -bimodule‎. ‎‎We show taht every Jordan higher derivation D=\{D_n\}_{n\in \mathbb{N}_۰} from the trivial extension \mathcal{A} \ltimes \mathcal{M} into itself is a higher derivation, if PD_۱(QXP)Q=QD_۱(PXQ)P=۰ for all X \in \mathcal{A} \ltimes \mathcal{M} ‎, in which P=(e,۰) and Q=(e^\prime,۰) for some non-trivial idempotent element e \in\mathcal{A} and e^\prime =۱_\mathcal{A}-e satisfying‎‎ ‎the following ‎conditions‎:‎‎‎e\mathcal{A}e^\prime\mathcal{A}e=\{۰\}‎‎, ‎‎e^\prime\mathcal{A}e\mathcal{A}e^\prime=\{۰\}‎‎‎,‎‎‎e(l.ann_\mathcal{A} \mathcal{M})e=\{۰\}‎‎‎, ‎‎e^\prime(r.ann_\mathcal{A} \mathcal{M})e^\prime=\{۰\}‎‎‎‎‎‎and eme^\prime=m for all m \in \mathcal{M} ‎.

کلیدواژه ها:

‎Jordan higher derivation‎ ، ‎Higher derivation‎ ، ‎Trivial extension‎ ، ‎Triangular algebra‎