# Divisibility Theory of Commutative Rings and Ideal Distributivity

## TL;DR

This paper explores a special class of commutative rings where elements have unique divisibility properties, establishing their algebraic structure and linking them to Bézout monoids through a representation theorem.

## Contribution

It characterizes a finitely axiomatizable class of rings with unique divisibility and connects their ideal structure to Bézout monoids via a new representation theorem.

## Key findings

- Class of rings is a finitely axiomatizable, ideal distributive quasivariety.
- This class is generated by integral domains with trivial units.
- Provides evidence supporting the conjecture linking Bézout monoids to ideal lattices.

## Abstract

We begin by investigating the class of commutative unital rings in which no two distinct elements divide the same elements. We prove that this class forms a finitely axiomatizable, relatively ideal distributive quasivariety, and it equals the quasivariety generated by the class of integral domains with trivial unit group. We end the paper by proving a representation theorem that provides more evidence to the conjecture that B\'ezout monoids describe exactly the monoids of finitely generated ideals of commutative unital rings with distributive ideal lattice.