On the atomicity of one-dimensional monoid algebras

TL;DR

This paper investigates the atomicity of monoid algebras over the rationals, confirming a conjecture that these algebras are atomic for a broad class of parameters, thus expanding understanding of atomic domains with specific algebraic properties.

Contribution

The authors prove that monoid algebras over the rationals are atomic for all parameters with odd denominators, advancing the verification of Gotti's conjecture.

Findings

Proved atomicity for monoid algebras with odd denominator parameters.

Extended previous results confirming atomicity for specific cases.

Contributed to the classification of atomic domains with Krull dimension one.

Abstract

The ascending chain condition on principal ideals (ACCP) is almost always complementary to atomicity within integral domains: in fact, Cohn initially stated that these two conditions were equivalent. This assertion has been shown to be false, however most counterexamples require technical algebraic constructions. In 2017, Gotti conjectured that for every $q$ in the set $S := ((0, 1) \cap \mathbb{Q}) \setminus {\mathbb{N}}^{-1}_{> 1}$, atomicity ascends from the exponentially cyclic Puiseux monoid $M_q$ to its monoid algebra over the field of rationals. If this conjecture were true, it would provide an extremely wide class of atomic domains of Krull dimension one not satisfying the ACCP, and so would be perhaps the simplest possible such examples. Bu et al. recently proved that the monoid algebra $\mathbb{Q} \left[M_{3/4} \right]$ is atomic, marking the first progress towards settling this conjecture. We strengthen this result and prove that $\mathbb{Q}[M_q]$ is atomic for all $q \in S$ having an odd denominator.