A Note on Posner's Theorem with Generalized Derivations on Lie Ideals

Abstract

Let R be a prime ring of characteristic different from 2, Z(R) its center, U its Utumi quotient ring, C its extended centroid,G a non-zero generalized derivation of R, L a non-centralLie ideal of R.We prove that if [[G(u),u],G(u)] ∈ Z(R) for all u ∈ L then one of the following holds:
1. there exists α ∈ C such that G(x) = α_x_, for all x ∈ R;
2. satisfies the standard identity _s_4(_x_1,…,x_4) and there exist a ∈ U, α ∈ C such that G(x) = ax + xa + α_x, for all x ∈ R.