Product of generalized skew derivations on Lie ideals

Let R be a non-commutative prime ring of characteristic different from 2, Qr be its right Martindale quotient ring and C be its extended centroid and L be a non-central Lie ideal of R Let F and G be two non-zero generalized skew derivations of R, associated with the same automorphism α and commuting with α. If (Formula presented.) for all (Formula presented.) then one of the following holds: there exist (Formula presented.) such that F(x) = ax and G(x) = cx, for all (Formula presented.) with ac = 0; (Formula presented.) the ring of 2 × 2 matrices over C, and there exist (Formula presented.) such that q is an invertible element of Qr, (Formula presented.) G(x) = cx, for all (Formula presented.) with (Formula presented.) (Formula presented.) and there exist (Formula presented.) and α automorphism of R, such that (Formula presented.) G(x) = cx, for all (Formula presented.) with (Formula presented.) Moreover, in this case α is not an inner automorphism of R. © 2021 Taylor & Francis Group, LLC.