Let R be a prime ring of characteristic different from 2 and 3, Qr its right Martindale quotient ring and C its extended centroid. Suppose that F, G are generalized skew derivations of R with the same associated automorphism α, and L is a non-central Lie ideal of R. If there exists an integer k ≥ 1 such that [F (x)x − xG(x), x]k = 0 for any x ∈ L, then 1. either there exist a ∈ Qr and λ ∈ C such that F (x) = xa and G(x) = (a + λ)x for all x ∈ R; 2. or there exists a field K such that R ⊆ M2 (K), the ring of 2 × 2 matrices over K, and one of the following conclusions occurs: (a) K is a finite field; (b) there exist a ∈ Qr and λ ∈ C such that F (x) + G(x) = ax + xa + λx for all x ∈ R. © 2023, Mathematical Society of the Rep. of China. All rights reserved.

### Generalized Skew Derivations with Hypercommuting Conditions on Lie Ideals

#### Abstract

Let R be a prime ring of characteristic different from 2 and 3, Qr its right Martindale quotient ring and C its extended centroid. Suppose that F, G are generalized skew derivations of R with the same associated automorphism α, and L is a non-central Lie ideal of R. If there exists an integer k ≥ 1 such that [F (x)x − xG(x), x]k = 0 for any x ∈ L, then 1. either there exist a ∈ Qr and λ ∈ C such that F (x) = xa and G(x) = (a + λ)x for all x ∈ R; 2. or there exists a field K such that R ⊆ M2 (K), the ring of 2 × 2 matrices over K, and one of the following conclusions occurs: (a) K is a finite field; (b) there exist a ∈ Qr and λ ∈ C such that F (x) + G(x) = ax + xa + λx for all x ∈ R. © 2023, Mathematical Society of the Rep. of China. All rights reserved.
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/11570/3284468`
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