Uniform approximation of functions with values in topological rings

Let (E, τ) be a non-Archimedean topological ring and let S be a zerodimensional compact Hausdorff topological space. We characterize the uniform closure of subsets of C(S; E) and obtain as corollaries the theorems of WeierstrassStone type of Dieudonné and Kaplansky, as well as their extension by Chernoff, Rasala and Waterhouse to a topological division ring whose topology is defined by a Krull valuation with arbitrary value group.