Product of generalized skew derivations on Lie ideals

Abstract

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 for all then one of the following holds: there exist such that F(x) = ax and G(x) = cx, for all with ac = 0; the ring of 2 × 2 matrices over C, and there exist such that q is an invertible element of Qr , G(x) = cx, for all with and there exist and α automorphism of R, such that G(x) = cx, for all with Moreover, in this case α is not an inner automorphism of R.