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]βZ(R) ([G(x),xβ]βZ(R)) for all xβR, then R is an order
in a central simple algebra of dimension at most 4 over its center or there
is an element Ξ» in the extended of R such that G(x)=Ξ»x
(G(x)=Ξ»βxβ) for all xβR