Annihilating and co-commuting conditions with generalized skew derivations

Let R be a prime ring of characteristic different from 2, Qr its right Martindale quotient ring and C its extended centroid. Suppose that F, G are generalized skew derivations of R associated with the same automorphism α of R, I anon-centralidealofRand0≠a∈R. Ifa (F(x)x−xG(x)) =0, forall x ∈ I, then there exist b,b′,q ∈ Qr and λ,θ ∈ C such that F(x) = bx+qxb′, G(x)=(θb′+λ)x, for any x∈R, with a(b−λ)=0 and a(q−θ)=0. In particular, if α is not an inner automorphism of R, then F (x) = bx, G(x) = λx, for any x ∈ R, with a(b − λ) = 0.