On a class of half-factorial domains

TL;DR

This paper investigates a specific property of integral domains called the Z-property, characterizes half-factorial polynomial rings under certain conditions, and explores implications for Krull domains with this property.

Contribution

It introduces and studies the Z-property, characterizes half-factorial polynomial rings with two-generated v-ideals, and links the property to torsion class groups in Krull domains.

Findings

Atomic domains with the Z-property are half-factorial.

Polynomial rings over domains with the Z-property are half-factorial under certain conditions.

Krull domains with the Z-property have torsion class groups.

Abstract

Let $R$ be an integral domain. For elements $a,b \in R$, let $[a,b]$ denote their greatest common divisor, if it exists. We say that $R$ has the Z-property if whenever $a,b,c,d$ and $e$ are nonzero nonunits of $R$ such that $abc=de$, then $[ab,d] \neq 1$ or $[ab,e] \neq 1$. The purpose of this paper is to study this property. The atomic integral domains that have this property constitute a class of half-factorial domains. Also, it is known that $R$ must have this property in order for the polynomial ring $R[x]$ to be half-factorial. We use it to give a characterization of half-factorial polynomial rings in the case where every $v$-ideal is $v$-generated by two elements. We also show that if $R$ is a Krull domain with this property, then $R$ has torsion class group.