- Hacettepe Journal of Mathematics and Statistics
- Vol: 52 Issue: 1
- On centrally extended Jordan derivations and related maps in rings
On centrally extended Jordan derivations and related maps in rings
Authors : Bharat Bhushan, Gurninder S. Sandhu, Shakir Ali, Deepak Kumar
Pages : 23-35
Doi:10.15672/hujms.1008922
View : 14 | Download : 4
Publication Date : 2023-02-15
Article Type : Research
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.Keywords : Prime ring, semiprime ring, centrally extended Jordan derivation, involution, centrally extended Jordan *-derivation