On centrally extended Jordan derivations and related maps in rings

Authors : Bharat BHUSHAN, Gurninder S SANDHU, Shakir ALİ, Deepak KUMAR

Pages : 23-35

Doi:10.15672/hujms.1008922

View : 55 | Download : 9

Publication Date : 2023-02-15

Article Type : Research Paper

Abstract :Let $R$ be a ring and $Zinsert ignore into journalissuearticles values(R);$ 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 insert ignore into journalissuearticles values(resp. centrally extended Jordan $\\ast$-derivation); $\\delta:R\\to R$ such that \\[ [\\deltainsert ignore into journalissuearticles values(x);,x]\\in Zinsert ignore into journalissuearticles values(R);~~insert ignore into journalissuearticles values(resp.~~[\\deltainsert ignore into journalissuearticles values(x);,x^{\\ast}]\\in Zinsert ignore into journalissuearticles values(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.
Keywords : Prime ring, semiprime ring, centrally extended Jordan derivation, involution, centrally extended Jordan derivation

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