Non-global nonlinear mixed skew Jordan Lie triple derivations on prime $ \ast $-rings

Fenhong Li; Liang Kong; Chao Li; Fenhong Li; Liang Kong; Chao Li

doi:10.3934/math.2025357

AIMS Mathematics

2025, Volume 10, Issue 4: 7795-7812. doi: 10.3934/math.2025357

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Research article

Non-global nonlinear mixed skew Jordan Lie triple derivations on prime $ \ast $-rings

School of Mathematics and Computer Application, Shangluo University, Shangluo 726000, China

Received: 04 December 2024 Revised: 07 March 2025 Accepted: 11 March 2025 Published: 02 April 2025
MSC : 16W10, 16W25

Let $ \mathcal{R} $ be a 2-torsion free unital prime $ \ast $-ring containing a nontrivial symmetric idempotent and $ Z_{c}(\mathcal {R}) $ be the anti-symmetric center of $ \mathcal {R} $. We prove that if a map $ \varphi:\mathcal{R}\rightarrow \mathcal{R} $ satisfies $ \varphi([A\bullet B, C]) = [\varphi(A)\bullet B, C]+[A\bullet\varphi(B), C] +[A\bullet B, \varphi(C)] $ for any $ A, B, C\in \mathcal{R} $ with $ ABC^{\ast} = 0 $, then there exists an additive $ \ast $-derivation $ \Theta $ of $ \mathcal{R} $ and a nonlinear map $ g:\mathcal{R}\rightarrow Z_{c}(\mathcal {R}) $ such that $ \varphi(A) = \Theta(A)+g(A) $ for any $ A\in \mathcal{R} $.
- mixed skew Jordan Lie triple derivation,
- additive $ \ast $-derivation,
- prime $ \ast $-ring
Citation: Fenhong Li, Liang Kong, Chao Li. Non-global nonlinear mixed skew Jordan Lie triple derivations on prime $ \ast $-rings[J]. AIMS Mathematics, 2025, 10(4): 7795-7812. doi: 10.3934/math.2025357

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Abstract

Let $ \mathcal{R} $ be a 2-torsion free unital prime $ \ast $-ring containing a nontrivial symmetric idempotent and $ Z_{c}(\mathcal {R}) $ be the anti-symmetric center of $ \mathcal {R} $. We prove that if a map $ \varphi:\mathcal{R}\rightarrow \mathcal{R} $ satisfies $ \varphi([A\bullet B, C]) = [\varphi(A)\bullet B, C]+[A\bullet\varphi(B), C] +[A\bullet B, \varphi(C)] $ for any $ A, B, C\in \mathcal{R} $ with $ ABC^{\ast} = 0 $, then there exists an additive $ \ast $-derivation $ \Theta $ of $ \mathcal{R} $ and a nonlinear map $ g:\mathcal{R}\rightarrow Z_{c}(\mathcal {R}) $ such that $ \varphi(A) = \Theta(A)+g(A) $ for any $ A\in \mathcal{R} $.

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