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.