On centrally-extended Jordan endomorophisms in rings

Abstract: The aim of this article is to introduce the concept of centrally-extended Jordan endomorphisms and proving that if $R$ is a non-commutative prime ring of characteristic not two, and $G$ is a CE- Jordan epimorphism such that $[G(x), x] \in Z(R)$ ($[G(x), x^*] \in Z(R)$) for all $x \in R$, then $R$ is an order in a central simple algebra of dimension at most $4$ over its center or there is an element $\lambda$ in the extended of $R$ such that $G(x) = \lambda x$ ($G(x) = \lambda^* x^*$) for all $x \in R$.