On finite factorization Puiseux algebras

TL;DR

This paper characterizes when Puiseux algebras over fields of characteristic zero have the finite factorization property, linking it to the monoid being a finite factorization monoid, and explores properties of generalized cyclotomic polynomials.

Contribution

It establishes that Puiseux algebras are FFD if and only if the underlying monoid is an FFM over fields of characteristic zero, extending understanding of factorization in monoid algebras.

Findings

Puiseux algebra $K[S]$ is an FFD iff $S$ is an FFM.

Every generalized cyclotomic polynomial has the finite factorization property in such algebras.

Provides a large class of one-dimensional monoid algebras with finite factorization property.

Abstract

An integral domain $D$ is called a finite factorization domain (FFD) if every nonzero nonunit element of $D$ has only finitely many non-associate divisors. In 1998, for an integral domain $D$ and a cancellative torsion-free monoid $S$ such that each nonzero element of its quotient group is of type $(0,0, \ldots)$, Kim proved that the monoid domain $D[S]$ is an FFD if and only if $D$ is an FFD and $S$ is an FFM. However, it is still open whether a monoid algebra $K[S]$ is an FFD provided that $S$ is a reduced FFM. In this paper, we show that a Puiseux algebra $K[S]$ is an FFD if and only if $S$ is an FFM, when $K$ is a finitely generated field of characteristic $0$. This would provide a large class of one-dimensional monoid algebras with finite factorization property. We also prove that every generalized cyclotomic polynomial has the finite factorization property in $K[S]$ where $S$ is a reduced FFM and $K$ is an arbitrary field of characteristic $0$.