One-dimensional monoid algebras and ascending chains of principal ideals

TL;DR

This paper constructs the first known one-dimensional monoid algebras that are atomic and satisfy the almost ACCP but do not satisfy the ACCP, providing elementary examples of such domains.

Contribution

It introduces the first one-dimensional monoid algebras satisfying the almost ACCP but not the ACCP, expanding the class of known atomic domains with specific chain conditions.

Findings

Constructed the first one-dimensional monoid algebras with these properties.

Provided elementary examples of atomic domains not satisfying the ACCP.

Extended understanding of chain conditions in monoid algebras.

Abstract

An integral domain $R$ is called atomic if every nonzero nonunit of $R$ factors into irreducibles, while $R$ satisfies the ascending chain condition on principal ideals if every ascending chain of principal ideals of $R$ stabilizes. It is well known and not hard to verify that if an integral domain satisfies the ACCP, then it must be atomic. The converse does not hold in general, but examples are hard to come by and most of them are the result of crafty and technical constructions. Sporadic constructions of such atomic domains have appeared in the literature in the last five decades, including the first example of a finite-dimensional atomic monoid algebra not satisfying the ACCP recently constructed by the second and third authors. Here we construct the first known one-dimensional monoid algebras satisfying the almost ACCP but not the ACCP (the almost ACCP is a notion weaker than the ACCP but still stronger than atomicity). Although the two constructions we provide here are rather technical, the corresponding monoid algebras are perhaps the most elementary known examples of atomic domains not satisfying the ACCP.