On Commutativity of Prime Rings With Generalized Derivations

Let R be an associative prime ring, U a Lie ideal such that u^2∈U for all u∈U and F: R→R be a generalized derivation. In this paper, we show that U ⊆ Z(R) if any one of the following conditions holds: (i) F(uv) - uv∈Z(R),(ii) F(uv) - vu∈Z(R), (iii) uF(v) + uv∈Z(R), and (iv) [F(u), v] + uv∈Z(R) for all u, v∈U. If we choose the underlying subset of R as an ideal instead of a Lie ideal of R, then we prove the commutativity of prime ring.