The number of atoms in an atomic domain

TL;DR

This paper investigates the structure of atomic domains with finitely many atoms and no prime elements, constructing examples with specific numbers of atoms and maximal ideals using algebraic and number theoretic methods.

Contribution

It provides explicit constructions of atomic domains with prescribed numbers of atoms and maximal ideals, expanding understanding of their possible configurations.

Findings

Existence of atomic domains with given atoms and maximal ideals

Construction methods combining algebra and number theory

Domains with no prime elements and specific atom counts

Abstract

We study the number of atoms and maximal ideals in an atomic domain with finitely many atoms and no prime elements. We show in particular that for all $m,n \in \mathbb{Z}^+$ with $n \geq 3$ and $4 \leq m \leq \frac{n}{3}$ there is an atomic domain with precisely $n$ atoms, precisely $m$ maximal ideals and no prime elements. The proofs use both commutative algebra and additive number theory.