Commutativity theorems for prime rings with generalized derivations on Jordan ideals

Abstract

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.