JORDAN HIGHER DERIVATIONS, A NEW APPROACH

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

References

[1] D. Benkovič, Jordan derivations and antiderivations on triangular matrices, Linear Algebra Appl., 397 (2005), 235–244.
[2] E. Christensen, Derivations of nest algebras, Math. Ann., 229(2) (1977), 155– 161.
[3] H. G. Dales, Banach Algebra and Automatic Continuity, Oxford: Oxford University Press. 2001.
[4] K. R. Davidson, Nest Algebras, Pitman research notes in mathematics series 191, Longman Sci. Tech., Harlow, 1988.
[5] H. R. Ebrahimi Vishki, M. Mirzavaziri and F. Moafian, Jordan higher derivations on trivial extension algebras, Commun. Korean Math. Soc., 31(2) (2016), 247–259.
[6] A. Erfanian Attar, H. R. Ebrahimi Vishki, Jordan derivations on trivial extension algebras, J. Adv. Res. Pure Math., 6(4) (2014), 24–32.
[7] M. Ferrero and C. Haetinger, Higher derivations and a theorem by Herstein, Quaest. Math., 25 (2002), 249–257.
[8] H. Ghahramani, Jordan Derivations on trivial extensions, Bull. Iranian Math. Soc., 39(4) (2013), 635–645.
[9] H. Hasse and F. K. Schmidt, Noch eine Begrüdung der theorie der höheren Differential quotienten in einem algebraischen Funtionenkörper einer Unbestimmeten, J. Reine Angew. Math., 177 (1937), 215–237.
[10] J. H. Zhang, W.Y. Yu, Jordan derivations of triangular algebras, Linear Algebra Appl., 419 (2006), 251–255.
Journal of Algebraic Systems
Volume 10, Issue 1
September 2022
Pages 167-177

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