Generalized derivations of left faithful rings

Let R be a left faithful ringU its right Utumi quotient ring and ρ a dense right ideal of R. An additive map g: ρ → U is called a generalized derivation if there exists a derivation δ of ρ into U such that for all x,y∈ρ. In this note, we prove that there exists an element a∈ U such that for all x ∈ ρ. From this characterization, it is proved that if R is a semiprime ring and if g is a generalized derivation with nilpotent values of bounded index, then g = 0. Analogous results are also obtained for the case of generalized derivations with nilpotent values on Lie ideals or one-sided ideals.