A note on characterization of higher derivations and their product

Document Type : Research Paper

Author

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

Abstract

‎There exists a one to one correspondence between higher derivations $\{d_n\}_{n=0}^\infty$ on an algebra $\mathcal{A}$ and the family of sequences of derivations $\{\delta_n\}_{n=1}^\infty$ on $\mathcal{A}$‎. ‎In this paper‎, ‎we obtain a relation that calculates each derivation $ \delta_n (n \in \mathbb{N})$ directly as a linear combination of products of terms of the corresponding higher derivation $\{d_n\}_{n=0}^\infty$‎. ‎Also‎, ‎we find the general form of the family of inner derivations corresponding to an inner higher derivation‎. ‎We show that for every two higher derivations on an algebra $\mathcal{A}$‎, ‎the product of them‎, ‎is a higher derivation on $\mathcal{A}$‎. ‎Also‎, ‎we prove that the product of two inner higher derivations‎, ‎is an inner higher derivation‎.

Keywords

Main Subjects

References

[1] Cortes, W., & Haetinger, C. (2005). On Jordan generalized higher derivations in rings. Turk. J. Math., 29, 1{10.
[2] Ekrami, S. Kh. (2022). Approximate orthogonally higher ring derivations. Control Optimization App. Math., 7(1), 93{106. doi: 10.30473/coam.2021.59727.1166
[3] Ekrami, S. Kh. Characterization of Hilbert C-module Higher Derivations. Georgian Mathematical Journal, accepted.
[4] Ekrami, S. Kh. (2022). Jordan higher derivations, a new approach. Journal Algebraic Systems, 10(1), 167{177.
[5] Haetinger, C. (2002). Higher derivations on Lie ideals. Tendencias em Matematica Aplicada e Computacional, 3, 141{145.
[6] Hasse, H., & Schmidt, F. K. (1937). Noch eine Begrudung der theorie der hoheren Di erential quotienten in einem algebraischen Funtionenkorper einer Unbestimmeten. J. Reine Angew. Math., 177, 215-237.
[7] Jewell, N. P. (1977). Continuity of module and higher derivations. Paci c J. Math., 68, 91{98.
[8] Johnson, B. E. (2001). Local derivations on C-algebras are derivations. Trans. Amer. Math. Soc., 353, 313{325.
[9] Johnson, B. E., & Sinclair, A. M. (1968). Continuity of derivations and a problem of Kaplansky. Amer. J. Math., 90, 1067{1073.
[10] Loy, R. J. (1973). Continuity of higher derivations. Proc. Amer. Math. Soc., 5, 505{510.
[11] Mirzavaziri, M. (2010), Characterization of higher derivations on algebras. Comm. Algebra, 38, 981{987.
[12] Mirzavaziri, M., Naranjani, K., & Niknam, A. (2010). Innerness of higher derivations. Banach J. Math. Anal., 4(2), 99{110.  DOI: 10.15352/bjma/1297117246
[13] Nowicki, A. (1984). Inner derivations of higher orders. Tsukuba J. Math., 8, 219{225.
[14] Roy, A., & Sridharan, R. (1968). Higher derivations and central simple algebras. Nagoya Math. J., 32, 21{30.
[15] Xu, S., & Xiao, Z. (2014). Jordan higher derivation revisited. Gulf J. Math., 2(1), 11{21.
Journal of Mahani Mathematical Research
Volume 13, Issue 1 - Serial Number 26
November 2023
Pages 403-415

Files

History

  • Receive Date: 20 April 2023
  • Revise Date: 30 August 2023
  • Accept Date: 06 October 2023

Share

How to cite

Statistics