Document Type : Original Manuscript

Author

Department of Mathematics, Payame Noor University, P.O. Box 19395-3697, Tehran, Iran.

Abstract

‎Let $\mathcal{A}$ be a unital algebra over a 2-torsion free commutative ring $\mathcal{R}$ and $\mathcal{M}$ be a unital $\mathcal{A}$-bimodule‎. ‎‎We show taht every Jordan higher derivation $D=\{D_n\}_{n\in \mathbb{N}_0}$ from the trivial extension $\mathcal{A} \ltimes \mathcal{M}$ into itself is a higher derivation, if $PD_1(QXP)Q=QD_1(PXQ)P=0$ for all $X \in \mathcal{A} \ltimes \mathcal{M}$‎, in which $P=(e,0)$ and $Q=(e^\prime,0)$ for some non-trivial idempotent element $e \in\mathcal{A}$ and $e^\prime =1_\mathcal{A}-e$ satisfying‎‎ ‎the following ‎conditions‎:
‎$‎‎e\mathcal{A}e^\prime\mathcal{A}e=\{0\}‎$‎, ‎$‎e^\prime\mathcal{A}e\mathcal{A}e^\prime=\{0\}‎$‎‎,
‎$‎‎e(l.ann_\mathcal{A} \mathcal{M})e=\{0\}‎$‎‎, ‎$‎e^\prime(r.ann_\mathcal{A} \mathcal{M})e^\prime=\{0\}‎$‎‎‎‎
‎and $eme^\prime=m$ for all $m \in \mathcal{M}$‎.

Keywords

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