Nonlinear $ \ast $-commuting maps on rings with involution

Yan Yang; Yan Yang

doi:10.3934/math.2026957

AIMS Mathematics

2026, Volume 11, Issue 8: 23782-23791. doi: 10.3934/math.2026957

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Research article

Nonlinear $ \ast $-commuting maps on rings with involution

Yan Yang ^{1,2
,
,}

1.
School of Mathematics and Computer Application, Shangluo University, Shangluo, 726000, China
2.
Key Laboratory of Intelligent Computing and Data Fusion, Shaanxi Provincial Institutions of Higher Education, Shangluo, 726000, China

Received: 20 March 2026 Revised: 21 May 2026 Accepted: 12 June 2026 Published: 03 August 2026
MSC : 16N60, 16W10

Let $ \mathcal{R} $ be a $ \ast $-ring with the unit $ I $. Assume that $ \mathcal{R} $ contains a symmetric idempotent $ P $ satisfying $ A\mathcal{R}P = 0 $ implies $ A = 0 $ and $ A\mathcal{R}(I-P) = 0 $ implies $ A = 0 $. We prove that if $ \varphi: \mathcal{R}\rightarrow \mathcal{R} $ is a nonlinear $ \ast $-commuting map, then there exist $ Z\in \mathcal{Z(R)} $ and a map $ h: \mathcal{R}\rightarrow \mathcal{Z(R)} $ such that $ \varphi(A) = ZA^{\ast}+h(A) $ for all $ A\in\mathcal{R} $.
- $ \ast $-commuting map,
- $ \ast $-ring,
- $ \ast $-Lie product,
- prime $ \ast- $ring,
- factor von Neumann algebra
Citation: Yan Yang. Nonlinear $ \ast $-commuting maps on rings with involution[J]. AIMS Mathematics, 2026, 11(8): 23782-23791. doi: 10.3934/math.2026957

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Abstract

Let $ \mathcal{R} $ be a $ \ast $-ring with the unit $ I $. Assume that $ \mathcal{R} $ contains a symmetric idempotent $ P $ satisfying $ A\mathcal{R}P = 0 $ implies $ A = 0 $ and $ A\mathcal{R}(I-P) = 0 $ implies $ A = 0 $. We prove that if $ \varphi: \mathcal{R}\rightarrow \mathcal{R} $ is a nonlinear $ \ast $-commuting map, then there exist $ Z\in \mathcal{Z(R)} $ and a map $ h: \mathcal{R}\rightarrow \mathcal{Z(R)} $ such that $ \varphi(A) = ZA^{\ast}+h(A) $ for all $ A\in\mathcal{R} $.

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