On Divisor Topology of Commutative Rings

TL;DR

This paper introduces a divisor topology on the set of equivalence classes of nonzero nonunits in an integral domain and explores its topological properties and connections to algebraic structures, providing new insights and proofs.

Contribution

It defines the divisor topology on $EC(R)$ and investigates its properties, linking algebraic features of $R$ with topological characteristics, including a new proof of the infinitude of primes.

Findings

Divisor topology forms a basis for $EC(R)$.

Characterization of valuation domains via nested properties.

New topological proof of infinitely many primes.

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 $D(R)$.