On the direct product of fields with an application

TL;DR

This paper investigates the properties of infinite direct products of fields, characterizes finiteness through ring theory, studies a specific localization, and relates set size to affine schemes.

Contribution

It introduces new characterizations of finiteness in direct products of fields and connects set size with scheme structure in algebraic geometry.

Findings

Finiteness characterized via ring-theoretic properties

Localization with cofinite sets studied

Finite sets correspond to affine schemes

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.