The Macias topology on integral domains

TL;DR

This paper generalizes a topology on positive integers to integral domains, exploring its properties and providing a new topological proof for the infinitude of prime elements, distinct from classical methods.

Contribution

It introduces a novel topology on integral domains inspired by the Macias topology and studies its properties, including a new proof of prime infinitude.

Findings

Topology properties on integral domains are characterized.

A new topological proof of the infinitude of primes is provided.

The topology's behavior on non-field principal ideal domains is analyzed.

Abstract

In this manuscript a recent topology on the positive integers generated by the collection of $\{\sigma_n:n\in\mathbb{N}\}$ where $\sigma_n:=\{m: \gcd(n,m)=1\}$ is generalized over integral domains. Some of its topological properties are studied. Properties of this topology on infinite principal ideal domains that are not fields are also explored, and a new topological proof of the infinitude of prime elements is obtained (assuming the set of units is finite or not open), different from those presented in the style of H. Furstenberg. Finally, some problems are proposed.