Valuation Ideal Factorization Domains

Abstract

An integral domain $D$ is a {\em valuation ideal factorization domain} (VIFD) if each nonzero principal ideal of $D$ can be written as a finite product of valuation ideals. Clearly, $\ddot{I}€$-domains are VIFDs. We study the ring-theoretic properties of VIFDs and the $\ast$-operation analogs of VIFDs. Among them, we show that if $D$ is treed (resp., $\ast$-treed), then $D$ is a VIFD (resp., $\ast$-VIFD) if and only if $D$ is an $\mathrm{h}$-local Prüfer domain (resp., a $\ast$-$\mathrm{h}$-local P$\ast$MD) if and only if every nonzero prime ideal of $D$ contains an invertible (resp., a $\ast$-invertible) valuation ideal. We also study integral domains $D$ such that for each nonzero nonunit $a\in D$, there is a positive integer $n$ such that $a^n$ can be written as a finite product of valuation elements.