Generalized Derivations via σ -Prime Rings

Abstract

Let Rbe a 2-torsion free σ-prime ring with involutionσ , Ua nonzero σ-Lie ideal, I a nonzero σ-ideal of R, S an appropriate subset of R and G a generalized derivation associated with a nonzero derivation δ of R commuting with σ. It is shown that the commutativity of a σ-prime ring R admitting a generalized derivation G satisfying one of the conditions: (P_1)[∂(x),G(y)]=±[x,y] and (P_2) ∂(x)∘G(y)=±x∘y, for everyx,y∈S.