Let R be a non-commutative prime ring of characteristic different from 2, U the Utumi quotient ring of R, C the extended centroid of R, L a non-central Lie ideal of R, G a non-zero generalized derivation of R.If [G(u), u](1) R satisfies the standard identity s(2) there exists γ ∈ C such that G(x) = γx for all x ∈ R.
Let R be a prime ring, with no nonzero nil right ideal, Q the two-sided Martindale quotient ring of R, F a generalized derivation of R, L a noncommutative Lie ideal of R, and b ∈ Q. If, for any u w ∈ L, there exists n = n u w ≥ 1 such that F uw − bwu n = 0, then one of the following statements holds:(a) F = 0 and b = 0; (b) R ⊆ M 2 K , the ring of 2 × 2 matrices over a field K, b 2 = 0, and F x = −bx, for all x ∈ R.
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