# On the atomicity of monoid algebras

## TL;DR

This paper demonstrates that even when both a monoid and a field are atomic, their monoid algebra may not be atomic, providing counterexamples across various ranks and characteristics.

## Contribution

It provides the first negative answers to the open question about atomicity of monoid algebras over fields when the monoid is atomic, with explicit constructions.

## Key findings

- Constructed atomic monoids of arbitrary infinite rank with non-atomic monoid algebras over any integral domain.
- Exhibited atomic monoids of finite rank over fields of finite characteristic with non-atomic monoid algebras.
- Built a rank 1 atomic monoid over the field with two elements where the monoid algebra is not atomic.

## Abstract

Let $M$ be a commutative cancellative monoid, and let $R$ be an integral domain. The question of whether the monoid ring $R[x;M]$ is atomic provided that both $M$ and $R$ are atomic dates back to the 1980s. In 1993, Roitman gave a negative answer to the question for $M = \mathbb{N}_0$: he constructed an atomic integral domain $R$ such that the polynomial ring $R[x]$ is not atomic. However, the question of whether a monoid algebra $F[x;M]$ over a field $F$ is atomic provided that $M$ is atomic has been open since then. Here we offer a negative answer to this question. First, we find for any infinite cardinal $\kappa$ a torsion-free atomic monoid $M$ of rank $\kappa$ satisfying that the monoid domain $R[x;M]$ is not atomic for any integral domain $R$. Then for every $n \ge 2$ and for each field $F$ of finite characteristic we exhibit a torsion-free atomic monoid of rank $n$ such that $F[x;M]$ is not atomic. Finally, we construct a torsion-free atomic monoid $M$ of rank $1$ such that $\mathbb{Z}_2[x;M]$ is not atomic.