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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