On finite factorization Puiseux algebras

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 $0$.