Posner’s second theorem and some related annihilating conditions on lie ideals

Let R be a non-commutative prime ring of characteristic different from 2 with Utumi quotient ring U and extended centroid C, L a non-central Lie ideal of R, F and G two non-zero generalized derivations of R. If [F(u), u]G(u) = 0 for all u ∈ L, then one of the following holds: (a) there exists λ ∈ C such that F(x) = λx, for all x ∈ R; (b) R ⊆ M2 (F), the ring of 2 × 2 matrices over a field F, and there exist a ∈ U and λ ∈ C such that F(x) = ax + xa + λx, for all x ∈ R. © 2018, University of Nis. All rights reserved.