# Irreducibility and factorizations in monoid rings

## TL;DR

This paper extends classical irreducibility tools to monoid rings and characterizes their algebraic properties, such as being Dedekind domains or UFDs, with a focus on atomicity and factorization behavior.

## Contribution

It generalizes Gauss's Lemma and Eisenstein's Criterion to monoid rings and classifies monoid algebras over nonnegative rationals by their algebraic structure.

## Key findings

- Extended Gauss's Lemma and Eisenstein's Criterion to monoid rings.
- Characterized when monoid algebras are Dedekind, Euclidean, PIDs, UFDs, or HFDs.
- Provided a characterization of submonoids of nonnegative rationals with half-factoriality.

## Abstract

For an integral domain $R$ and a commutative cancellative monoid $M$, the ring consisting of all polynomial expressions with coefficients in $R$ and exponents in $M$ is called the monoid ring of $M$ over $R$. An integral domain is called atomic if every nonzero nonunit element can be written as a product of irreducibles. In the investigation of the atomicity of integral domains, the building blocks are the irreducible elements. Thus, tools to prove irreducibility are crucial to study atomicity. In the first part of this paper, we extend Gauss's Lemma and Eisenstein's Criterion from polynomial rings to monoid rings. An integral domain $R$ is called half-factorial (or an HFD) if any two factorizations of a nonzero nonunit element of $R$ have the same number of irreducible elements (counting repetitions). In the second part of this paper, we determine which monoid algebras with nonnegative rational exponents are Dedekind domains, Euclidean domains, PIDs, UFDs, and HFDs. As a side result, we characterize the submonoids of $(\mathbb{Q}_{\ge 0},+)$ satisfying a dual notion of half-factoriality known as other-half-factoriality.