Abstract
We study the notions of centrally-extended higher \(*\)-derivations and centrally-extended generalized higher \(*\)-derivations. Both are shown to be additive in a \(*\)-ring without nonzero central ideals. Also, we prove that in semiprime \(*\)-rings with no nonzero central ideals, every centrally-extended (generalized) higher \(*\)-derivation is a (generalized) higher \(*\)-derivation.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Data Availability
Not applicable.
Code Availability
Not applicable.
References
Ashraf, M., Parveen, N.: Lie triple higher derivable maps on rings. Commun. Algebra 45(5), 2256–2275 (2017)
Bell, H.E., Daif, M.N.: On centrally-extended maps on rings. Beitr. Algebra Geom. 57, 129–136 (2016)
Brešar, M., Vukman, J.: On some additive mappings in rings with involution. Aequat. Math. J. 38, 178–185 (1989)
Daif, M.N., Tammam El-Sayiad, M.S.: On Jordan and Jordan \(\ast \)-generalized derivations in semiprime rings with involution. Int. J. Contemp. Math. Sci. 2(30), 1487–1492 (2007)
El-Deken, S.F., Nabiel, H.: Centrally-extended generalized \(^\ast \)-derivations on rings with involution. Beitr. Algebra Geom. 60, 217–224 (2019)
Ezzat, O.H.: A note on Jordan triple higher \(\ast \)-derivations on semiprime rings. Int. Sch. Res. Not. 2014, 5 (2014)
Ezzat, O.H.: Generalized Jordan triple higher \(\ast \)-derivaions on semiprime rings. Gen. Math. Notes 23(2), 1–10 (2014)
Ezzat, O.H.: Functional equations related to higher derivations in semiprime rings. Open Math. 19(1), 1359–1365 (2021). https://doi.org/10.1515/math-2021-0123
Ferrero, M., Haetinger, C.: Higher derivations of semiprime rings. Commun. Algebra 30(5), 2321–2333 (2002)
Ferrero, M., Haetinger, C.: Higher derivations and a theorem by Herstein. Quaest. Math. 25(2), 249–257 (2002)
Fošner, A.: Generalized higher derivations on algebras. Ukrain. Math. J. 69(10), 1659–1667 (2018)
Mirzavaziri, M.: Characterization of higher derivations on algebras. Commun. Algebra 38(3), 981–987 (2010)
Wei, F., Xiao, Z.: Higher derivations of triangular algebras and its generalizations. Linear Algebra Appl. 435(5), 1034–1054 (2011)
Zalar, B.: On centralizers of semiprime rings. Comment. Math. Univ. 32(4), 609–614 (1991)
Funding
This research received no external funding.
Author information
Authors and Affiliations
Contributions
One author.
Corresponding author
Ethics declarations
Conflict of interest
The author declares that there is no potential conflict of Interest regarding publishing this paper.
Additional information
Communicated by Uwe Kaehler.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Ezzat, O.H. Rings with Centrally-Extended Higher \(*\)-Derivations. Adv. Appl. Clifford Algebras 33, 21 (2023). https://doi.org/10.1007/s00006-023-01265-z
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00006-023-01265-z