On the graph of divisibility of an integral domain

TL;DR

This paper introduces a topological and graph-theoretic perspective to analyze the factorization properties of integral domains, linking connectedness in these structures to generalized atomicity.

Contribution

It presents a novel approach by applying topology and graph theory to the study of divisibility in integral domains, extending the understanding of atomicity.

Findings

Connectedness in the graph correlates with atomicity properties.

Topological structures provide new insights into factorization behavior.

Generalization of atomicity through topological and graph-theoretic concepts.

Abstract

It is well-known that the factorization properties of a domain are reflected in the structure of its group of divisibility. The main theme of this paper is to introduce a topological/graph-theoretic point of view to the current understanding of factorization in integral domains. We also show that connectedness properties in the graph and topological space give rise to a generalization of atomicity.