ON CENTRALLY EXTENDED JORDAN DERIVATIONS AND RELATED MAPS IN RINGS

Author:

BHUSHAN Bharat1,SANDHU Gurninder S.2,ALİ Shakir3,KUMAR Deepak1

Affiliation:

1. Punjabi University

2. Patel Memorial National College

3. Aligarh Muslim University

Abstract

Let $R$ be a ring and $Z(R)$ be the center of $R.$ The aim of this paper is to define the notions of centrally extended Jordan derivations and centrally extended Jordan $\ast$-derivations, and to prove some results involving these mappings. Precisely, we prove that if a $2$-torsion free noncommutative prime ring $R$ admits a centrally extended Jordan derivation (resp. centrally extended Jordan $\ast$-derivation) $\delta:R\to R$ such that \[ [\delta(x),x]\in Z(R)~~(resp.~~[\delta(x),x^{\ast}]\in Z(R))\text{~for~all~}x\in R, \] where $'\ast'$ is an involution on $R,$ then $R$ is an order in a central simple algebra of dimension at most 4 over its center.

Publisher

Hacettepe University

Subject

Geometry and Topology,Statistics and Probability,Algebra and Number Theory,Analysis

Reference27 articles.

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2. \bibitem{Ali2013} {\sc S. Ali, A. Fo$\check{s}$ner, Maja Fo$\check{s}$ner, M. S. Khan}, {\sl On generalized Jordan triple $(\alpha,\beta)^{\ast}$-derivations and related mappings}, Mediterr. J. Math., {\bf 10} (2013), 1657-1668.

3. \bibitem{Ashraf2000} {\sc M. Ashraf, N. Rehman}, {\sl On Jordan generalized derivations in rings}, Math. J. Okayama Univ., {\bf 42} (2000), 7–9.

4. \bibitem{Ashraf2006} {\sc M. Ashraf, S. Ali, C. Haetinger}, {\sl On derivations in rings and their applications}, The Aligarh Bull. Math., {\bf 25 (2)} (2006), 79-107.

5. \bibitem{Beidar1996} {\sc K. I. Beidar, W. S. Martindale III, A. V. Mikhalev}, {\sl Rings with Generalized Identities}, Pure Appl. Math. {\bf 196}, Marcel Dekker Inc., New York, 1996.

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2. Multiplicative Generalized Reverse CE-Derivations Acting on Rings with Involution;Journal of Mathematics;2023-05-24

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4. Centrally-extended generalized Jordan derivations in rings;Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica;2023-01-01

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