Nonlinear $ \ast $-commuting maps on rings with involution
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} $.
Fields of Study & Subjects
Advanced Topics in Algebra
Rings
Modules
and Algebras
Advanced Operator Algebra Research
Publication details
- Journal / source
- AIMS Mathematics
- Publisher
- American Institute of Mathematical Sciences (AIMS)
- Volume
- 11
- Issue
- 8
- Pages
- 23782–23791
- Publication date
- 1 January 2026