  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.
