Maximal common divisors in monoid algebras

Abstract

A commutative cancellative monoid has the MCD property if every nonempty finite subset has a maximal common divisor, and it has the MCD-finite property if every such subset has only finitely many maximal common divisors up to associates. We study these properties in monoid algebras. We first identify atomic and antimatter families of rank-$1$ torsion-free MCD monoids. We then prove that, for a field F and a torsion-free monoid M, the MCD property ascends from M to F[M] whenever the group algebra F[𝒢(M)] satisfies the ascending chain condition on principal ideals (𝒢(M) denotes the Grothendieck group of M). We also record the pre-Schreier case as a consequence of classical GCD ascent. In contrast, for every prime p, we construct a reduced rank-$3$ MCD monoid M such that 𝔽_p[M] is not an MCD domain, and an atomic rank-$1$ MCD monoid whose algebra over 𝔽_p is not atomic. We further prove that the prime reciprocal monoids considered here are MCD-finite and that MCD-finiteness ascends to polynomial extensions. Finally, for every prime p, we construct a rank-$1$ torsion-free q-GCD monoid whose algebra over 𝔽_p is not q-GCD, where the q-GCD property requires at most one maximal common divisor up to associates for every nonempty finite subset.

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BibTeX RIS

BibTeX

@misc{blitz2026maximal,
  title = {{Maximal common divisors in monoid algebras}},
  author = {Grant Blitz and Felix Gotti and Darren Han and Hengrui Liang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2609.18947},
  eprint = {2609.18947},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.18947},
}

RIS

TY  - UNPB
TI  - Maximal common divisors in monoid algebras
AU  - Grant Blitz
AU  - Felix Gotti
AU  - Darren Han
AU  - Hengrui Liang
PY  - 2026
DA  - 2026/10/21
PB  - arXiv
DO  - 10.48550/arxiv.2609.18947
UR  - https://arxiv.org/abs/2609.18947
AB  - A commutative cancellative monoid has the MCD property if every nonempty finite subset has a maximal common divisor, and it has the MCD-finite property if every such subset has only finitely many maximal common divisors up to associates. We study these properties in monoid algebras. We first identify atomic and antimatter families of rank-$1$ torsion-free MCD monoids. We then prove that, for a field F and a torsion-free monoid M, the MCD property ascends from M to F[M] whenever the group algebra F[𝒢(M)] satisfies the ascending chain condition on principal ideals (𝒢(M) denotes the Grothendieck group of M). We also record the pre-Schreier case as a consequence of classical GCD ascent. In contrast, for every prime p, we construct a reduced rank-$3$ MCD monoid M such that 𝔽_p[M] is not an MCD domain, and an atomic rank-$1$ MCD monoid whose algebra over 𝔽_p is not atomic. We further prove that the prime reciprocal monoids considered here are MCD-finite and that MCD-finiteness ascends to polynomial extensions. Finally, for every prime p, we construct a rank-$1$ torsion-free q-GCD monoid whose algebra over 𝔽_p is not q-GCD, where the q-GCD property requires at most one maximal common divisor up to associates for every nonempty finite subset.
ER  -

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