Some Topology on Zero-Dimensional Subrings of Product of Rings

Abstract

Let $R$ be a ring and $\{R_{i}\}_{i\in I}$ a family of zero-dimensional rings. We define the Zariski topology on $\mathcal{Z}(R,\prod R_{i})$ and study their basic properties. Moreover, we define a topology on $\mathcal{Z}(R,\prod R_{i})$ by using ultrafilters it's called the ultrafilter topology and we demonstrate that this topology is finer than the Zariski topology. We show that the ultrafilter limit point of a collections of subrings of $\mathcal{Z}(R,\prod R_{i})$ is a zero-dimensional ring. Thereby, its relationship with $\mathcal{F}-\lim$ and the direct limit of a family of
rings.