Generalized derivations on Jordan ideals in prime rings

Let R be a 2-torsion free prime ring with center Z(R) , J be a nonzero Jordan ideal also a subring of R , and F be a generalized derivation with associated derivation d. In the present paper, we shall show that J ⊆ Z(R) if any one of the following properties holds: (i) [F (u), u] ∈ Z(R) , (ii) F (u)u = ud(u) , (iii) d(u^2) = 2F (u)u , (iv) F (u^2) − 2uF (u) = d(u^2) − 2ud(u) , (v) F^2(u) + 3d^2(u) = 2F d(u) + 2dF (u) , (vi) F (u^2) = 2uF (u) for all u ∈ J.