Centrally-extended generalized Jordan derivations in rings (Q6167810)

Let \(R\) be an associative ring, \(Z(R)\) its center, \(C\) its extended centroid and \(*:R\mapsto R\) an involution of \(R\). For any \(x,y\in R\), the anti-commutator of \(x,y\) is defined to be \(x\circ y=xy+yx\). A map \(\delta: R \mapsto R\) is called \textit{centrally extended Jordan derivation} of \(R\), if \(\delta(x+y)-\delta(x)-\delta(y)\in Z(R)\) and \(\delta(x\circ y)-\delta(x)\circ y-x\circ \delta(y)\in Z(R)\), for all \(x,y \in R\). In the paper under review, the authors introduce a definition of a map generalizing the concept of centrally extended Jordan derivation. More precisely, a map \(F:R\mapsto R\) is said to be a \textit{centrally extended generalized Jordan derivation} of \(R\), constrained with a centrally Jordan derivation \(\delta\) of \(R\), if \(F(x+y)-F(x)-F(y)\in Z(R)\) and \(F(x\circ y)-F(x)y-F(y)x-x\delta(y)-y\delta(x)\in Z(R)\), for all \(x,y \in R\). The main objective of the paper is to investigate the structure of a non-commutative prime ring, under the hypothesis that a centrally extended generalized Jordan derivation is centralizing (or \(*\)-centralizing) on \(R\). More precisely, the authors prove the following results: \begin{itemize} \item[1.] (Theorem 1.2) Let \(R\) be a \(2\)-torsion free noncommutative prime ring, \(F:R\mapsto R\) a centrally extended generalized Jordan derivation of \(R\). If \([F(x),x]\in Z(R)\) for any \(x\in R\) (i.e., \(F\) is centralizing on \(R\)), then either \(R\) is an order in a central simple algebra of dimension at most \(4\) over its center or there exists \(\lambda \in C\) such that \(F(x)=\lambda x\), for any \(x\in R\). \item[2.] (Theorem 1.3) Let \(R\) be a \(2\)-torsion free noncommutative prime ring, \(F:R\mapsto R\) a centrally extended generalized Jordan derivation of \(R\). If \([F(x),x^*]\in Z(R)\) for any \(x\in R\) (i.e., \(F\) is \(*\)-centralizing on \(R\)), then either \(R\) is an order in a central simple algebra of dimension at most \(4\) over its center or \(F=0\). \end{itemize}