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

LetR be a prime ring with center Z(R), λ a nonzero left ideal, α, β are automorphisms ofR and R admits a generalized (α, β)-derivation F associated with a nonzero (α, β)-derivation d such that d(Z(R)) 6= (0). In the present paper, we prove that if any one of the following holds: (i) F ([x, y])− bα(x ◦ y) ∈ Z(R) (ii) F ([x, y]) + bα(x ◦ y) ∈ Z(R) (iii) F (x ◦ y)− bα([x, y]) ∈ Z(R) (iv) F (x ◦ y) + bα([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.