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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.

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