"Generalized (α,β)-derivations and left Ideals in Prime Rings"

Let R be a prime ring with center Z(R), λ a nonzero left ideal, α, β are automorphisms of R and R admits a generalized (α, β)-derivation F associated with a nonzero (α, β)-derivation d such that d(Z(R)) = (0).In the present paper, we prove that if any one of the following holds: ([x, y]) ∈ Z(R) for all x, y ∈ λ and for some b ∈ R then R is commutative.Also some related results have been obtained.