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

Abstract

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 ? ? C such that F(x) = ?x, for all x ? R; (b) R ? M2(F), the ring of 2 x 2 matrices over a field F, and there exist a ? U and ? ? C such that F(x) = ax + xa + ?x, for all x ? R.