On the atomicity of monoid algebras

By J. Coykendall, Felix Gotti. Journal of Algebra, 2019

59 citations recorded by the source corpus.

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.

Subfields: Algebra and Number Theory, Artificial Intelligence, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/c4fbd8cd6c4a0fac4b1246edfec0d26e202bd529

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In Algebra and Number Theory

  1. S. Tringali
  2. Felix Gotti
  3. Alfred Geroldinger

In Artificial Intelligence

  1. Alfred Geroldinger
  2. S. Tringali

In Geometry and Topology

  1. Alfred Geroldinger
  2. S. Tringali

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