Unique factorization property of non-unique factorization domains II

TL;DR

This paper investigates valuation factorization domains (VFDs), exploring their properties, relationships with other domain classes, and their unique factorization characteristics, thereby advancing understanding of non-unique factorization domains.

Contribution

It establishes key properties of VFDs, links them to P$v$MDs and GCD-domains, and characterizes their unique factorization behavior, providing new insights into their structure.

Findings

VFDs are Schreier domains with trivial class group

VFDs coincide with weakly Matlis GCD-domains in P$v$MDs

VFDs are archimedean if and only if they are weakly factorial GCD-domains

Abstract

Let $D$ be an integral domain. A nonzero nonunit $a$ of $D$ is called a valuation element if there is a valuation overring $V$ of $D$ such that $aV\cap D=aD$. We say that $D$ is a valuation factorization domain (VFD) if each nonzero nonunit of $D$ can be written as a finite product of valuation elements. In this paper, we study some ring-theoretic properties of VFDs. Among other things, we show that (i) a VFD $D$ is Schreier, and hence ${\rm Cl}_t(D)=\{0\}$, (ii) if $D$ is a P$v$MD, then $D$ is a VFD if and only if $D$ is a weakly Matlis GCD-domain, if and only if $D[X]$, the polynomial ring over $D$, is a VFD and (iii) a VFD $D$ is a weakly factorial GCD-domain if and only if $D$ is archimedean. We also study a unique factorization property of VFDs.