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 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 AM into itself is a higher derivation, if PD1(QXP)Q=QD1(PXQ)P=0 for all XAM‎, in which P=(e,0) and Q=(e,0) for some non-trivial idempotent element eA and e=1Ae satisfying‎‎ ‎the following ‎conditions‎:
eAeAe={0}‎, ‎eAeAe={0}‎‎,
e(l.annAM)e={0}‎‎, ‎e(r.annAM)e={0}‎‎‎‎
‎and eme=m for all mM‎.

Keywords


[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.