Nonlinear skew commuting maps on $\ast$-rings

  • L. Kong School Math. and Statistics, Shaanxi Normal Univ., and Inst. Appl. Math., Shangluo Univ., China
  • J. Zhang School Math. and Statistics, Shaanxi Normal Univ., China
Keywords: Commuting maps; skew commuting maps; rings

Abstract

UDC 512.5

Let $\mathcal{R}$ be a unital $\ast$-ring with the unit $I$. Assume that $\mathcal{R}$ contains a symmetric idempotent $P$ which satisfies $A{\mathcal{R}}P = 0$ implies $A=0$ and $A{\mathcal{R}}(I-P) = 0$ implies $A = 0$. In this paper, it is proved that if $\phi\colon\mathcal{R} \rightarrow \mathcal{R}$ is a nonlinear skew commuting map, then there exists an element $Z \in \mathcal{Z}_{S}(\mathcal{R})$ such that $\phi(X) = ZX$ for all $X \in \mathcal{R}$, where $\mathcal{Z}_{S}(\mathcal{R})$ is the symmetric center of $\mathcal{R}$.
As an application, the form of nonlinear skew commuting maps on factors is obtained.

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Published
07.07.2022
How to Cite
Kong, L., and J. Zhang. “Nonlinear Skew Commuting Maps on $\ast$-Rings”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 74, no. 6, July 2022, pp. 826 -31, doi:10.37863/umzh.v74i6.801.
Section
Research articles