Commuting maps on alternative rings

Abstract

Suppose $\mathfrak{R}$ is a $2$,$3$-torsion free unital alternative ring having an idempotent element $e_1$ $\left(e_2 = 1-e_1\right)$ which satisfies $x \mathfrak{R} \cdot e_i = \{0\} \rightarrow x = 0$ $\left(i = 1,2\right)$. In this paper, we aim to characterize the commuting maps. Let $\varphi$ be a commuting map of $\mathfrak{R}$ so it is shown that $\varphi(x) = zx + \Xi(x)$ for all $x \in \mathfrak{R}$, where $z \in \mathcal{Z}(\mathfrak{R})$ and $\Xi$ is an additive map from $\mathfrak{R}$ into $\mathcal{Z}(\mathfrak{R})$. As a consequence a characterization of anti-commuting maps is obtained and we provide as an application, a characterization of commuting maps on von Neumann algebras relative alternative $C^{*}$-algebra with no central summands of type $I_1$.