Commuting maps on alternative rings

Suppose $\mathfrak{R}$ is a $2$,$3$-torsion free unital alternative ring having an idempotent element $e_1$ $(e_2=1-e_1)$ which satisfies $x\mathfrak{R}\cdot e_i=\{0\}\to x=0$ $(i=1,2)$. In this paper, we aim to characterize the commuting maps. Let $\phi$ be a commuting map of $\mathfrak{R}$ so it is shown that $\phi (x)=zx+\mathrm{\Xi }(x)$ for all $x\in \mathfrak{R}$, where $z\in \mathcal{Z}(\mathfrak{R})$ and $\mathrm{\Xi }$ 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 $C^{\ast }$-algebra with no central summands of type $I_1$.