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arXiv · 2202.07459

On centrally extended Jordan derivations and related maps in rings

Bharat Bhushan·Gurninder Singh Sandhu·Shakir Ali·Deepak Kumar
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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.

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Bharat Bhushan, Gurninder Singh Sandhu, Shakir Ali, Deepak Kumar. 2022-01-25. On centrally extended Jordan derivations and related maps in rings. https://arxiv.org/abs/2202.07459

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