On Generalized (α, β)-Derivations of Rings with Involution

Let α, β be automorphisms of a semiprime ∗-ring R. We show that: (i) If F: R → R is an additive mapping, where R is a 2-torsion free semiprime ∗-ring and D an (α, β)-derivation of R, satisfying F (xx∗) = F (x)α(x∗) + β(x)D(x∗) for all x ∈ R, then F is a generalized (α, β)-derivation. (ii) If F: R → R is an additive mapping, where R is a 6-torsion free semiprime ∗-ring and D a Jordan triple (α, β)∗- derivation of R, satisfying F (xyx) = F (x)α(y∗x∗) + β(x)D(y)α(x∗) + β(xy)D(x) for all x, y ∈ R, then F is a Jordan triple generalized (α, β)∗-derivation. (iii) Let R be a 2-torsion free semiprime ∗−ring. If F: R → R is a left generalized (α, β)∗-derivation with associated left (α, β)∗-derivation D of R such that [F (x)+D(x), α(x∗)] = 0 for all x ∈ R, then D(x) ∈ Z(R).