Commutativity theorems for prime rings with generalized derivations on Jordan ideals

Let R be a 2-torsion free prime ring, J a nonzero Jordan ideal of R and F be a generalized derivation associated with a nonzero derivation d. If F satisfies any one of the following conditions: (i) F(xy) − xy ∈ Z(R); (ii) F(xy) − yx ∈ Z(R); (iii) F(x)F(y) − xy ∈ Z(R); (iv) F(x)F(y) − yx ∈ Z(R) for all x, y ∈ J, then R is commutative.