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arxiv: 1906.11138 · v1 · pith:IF4TKKVNnew · submitted 2019-06-26 · 🧮 math.AC

On the atomicity of monoid algebras

Jim Coykendall , Felix Gotti This is my paper

Pith reviewed 2026-05-25 14:59 UTC · model grok-4.3

classification 🧮 math.AC
keywords atomic monoidsmonoid algebrasmonoid ringsatomicitytorsion-free monoidscommutative algebrafactorization
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The pith

Atomic monoids of any rank can produce non-atomic monoid algebras over fields.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines whether an atomic cancellative monoid M guarantees that the monoid algebra R[x;M] is atomic whenever the base ring R is atomic. Previous work gave a negative answer when M is the non-negative integers, but left open the case of a field base. The authors resolve the question negatively by explicit constructions. They produce torsion-free atomic monoids M of every infinite rank such that R[x;M] fails to be atomic for any integral domain R. Parallel constructions give counterexamples of every finite rank at least two over fields of finite characteristic, and a rank-one counterexample over the field with two elements.

Core claim

For any infinite cardinal κ there exists a torsion-free atomic monoid M of rank κ such that the monoid domain R[x;M] is not atomic for any integral domain R. For every n ≥ 2 and every field F of finite characteristic there exists a torsion-free atomic monoid M of rank n such that F[x;M] is not atomic. There also exists a torsion-free atomic monoid M of rank 1 such that ℤ₂[x;M] is not atomic.

What carries the argument

Torsion-free atomic monoids of prescribed rank, built so that the associated monoid algebra contains non-atomic elements despite the monoid itself being atomic.

If this is right

  • Atomicity of M does not transfer to atomicity of R[x;M] when R is any integral domain and M has infinite rank.
  • Atomicity of M does not transfer to atomicity of F[x;M] when F has finite characteristic and M has finite rank at least two.
  • Atomicity of M does not transfer to atomicity of ℤ₂[x;M] even when M has rank one.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The characteristic of the base field appears to play a role in whether counterexamples of small rank exist.
  • The same monoid constructions may be examined for preservation or failure of other factorization properties such as being a unique factorization monoid.

Load-bearing premise

The paper's constructions succeed in producing monoids that remain atomic and torsion-free while the corresponding algebra becomes non-atomic.

What would settle it

An explicit factorization into atoms of every non-unit element inside one of the constructed algebras F[x;M] would show that the given monoid does not serve as a counterexample.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 1 minor

Summary. The manuscript constructs torsion-free atomic monoids M of specified ranks such that the monoid algebra R[x;M] fails to be atomic, providing negative answers to the long-standing question of whether atomicity of M implies atomicity of R[x;M] when R is an integral domain (or field). The three main results are: (i) for any infinite cardinal κ, an M of rank κ making R[x;M] non-atomic for every integral domain R; (ii) for each n≥2 and field F of finite characteristic, an M of rank n making F[x;M] non-atomic; (iii) a rank-1 M making ℤ₂[x;M] non-atomic.

Significance. If the explicit constructions are correct and the atomicity/non-atomicity properties are verified, the paper resolves an open question from the 1980s–1990s on atomicity transfer in monoid algebras over fields by supplying the first counterexamples. The graded constructions across infinite and finite ranks, including characteristic-dependent cases, constitute a substantive advance in the theory of atomic monoids and their algebras.

minor comments (1)
  1. The abstract asserts existence of the monoids with the stated properties; the body should include explicit verification steps (e.g., checking the atomicity condition on generators and the failure of atomicity in the algebra via specific non-factorable elements) for each of the three constructions to make the claims fully self-contained.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and for recommending acceptance. The referee's summary accurately captures the main results and their significance in resolving the open question on atomicity transfer for monoid algebras over fields.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper establishes negative answers to an open question via explicit constructions of torsion-free atomic monoids M (of various ranks) such that R[x;M] fails to be atomic. Atomicity, torsion-freeness, cancellativity, and rank are introduced via standard definitions in the area; the proofs consist of direct verification that the constructed objects satisfy the stated properties. No step reduces a claimed prediction or uniqueness result to a fitted parameter, self-citation chain, or definitional tautology. The central existence claims remain independent of any internal circular reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only information yields no identifiable free parameters, background axioms, or invented entities; the results are stated as existence of certain monoids.

pith-pipeline@v0.9.0 · 5769 in / 1092 out tokens · 33289 ms · 2026-05-25T14:59:32.068027+00:00 · methodology

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Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages

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