On Divisor Topology of Commutative Rings
Uğur Yiğit, Suat Koç
TL;DR
This work introduces the divisor topology $D(R)$ on the set of divisor-classes $EC(R^{\#})$ of an integral domain $R$, extending Steen's divisor topology on $\mathbb{Z}$ to general domains. It investigates how algebraic properties (e.g., valuation domains, gcd-domains, atomic domains) shape topological features of $D(R)$, establishing results such as $D(R)$ being Alexandrov, $T_0$ but not $T_1$ or Hausdorff, and characterizing valuation domains via nested basis structure. A key finding is that $D(R)$ is Noetherian if and only if $R$ is a field, with several equivalent formulations, and that $D(R)$ is ultraconnected and path-connected while failing to be regular or compact. The paper also uses $D(R)$ to provide a topological proof of the infinitude of primes in a UFD, illustrating the utility of the topology in classical number-theoretic contexts and highlighting the deep links between divisibility in rings and topological properties.
Abstract
Let $R\ $be an integral domain and $R^{\#}$ the set of all nonzero nonunits of $R.\ $For every elements $a,b\in R^{\#},$ we define $a\sim b$ if and only if $aR=bR,$ that is, $a$ and $b$ are associated elements. Suppose that $EC(R^{\#})$ is the set of all equivalence classes of $R^{\#}\ $according to $\sim$.$\ $Let $U_{a}=\{[b]\in EC(R^{\#}):b\ $divides $a\}$ for every $a\in R^{\#}.$ Then we prove that the family $\{U_{a}\}_{a\in R^{\#}}$ becomes a basis for a topology on $EC(R^{\#}).\ $This topology is called divisor topology of $R\ $and denoted by $D(R).\ $We investigate the connections between the algebraic properties of $R\ $and the topological properties of$\ D(R)$. In particular, we investigate the seperation axioms on $D(R)$, first and second countability axioms, connectivity and compactness on $D(R)$. We prove that for atomic domains $R,\ $the divisor topology $D(R)\ $is a Baire space. Also, we characterize valution domains $R$ in terms of nested property of $D(R).$ In the last section, we introduce a new topological proof of the infinitude of prime elements in a UFD and integers by using the topology $D(R)$.