On the direct product of fields with an application

Abstract

In this paper, the (infinite) direct product of fields is investigated. In particular, the finiteness of a given set is characterized in terms of some ring-theoretic observations. Next, a certain localization (whose multiplicative set formed by cofinite sets) of the direct product of fields is studied. Finally, it is shown that every set $X$ can be made into a separated scheme, and this scheme is an affine scheme if and only if $X$ is a finite set.