On centrally-extended Jordan endomorophisms in rings

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 $[G(x),x]\in Z(R)$ ($[G(x),x^{\ast }]\in Z(R)$) 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 $G(x)=\lambda x$ ($G(x)={\lambda }^{\ast }{x}^{\ast }$) for all $x\in R$.