On centrally extended Jordan derivations and related maps in rings

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 *-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 *-derivation) 6 : R -> R such that[6(x), x] E Z(R) (resp. [6(x), x*] E Z(R)) for all x E R,where '*' is an involution on R, then R is an order in a central simple algebra of dimension at most 4 over its center.