Valuation Ideal Factorization Domains

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, $\pi$-domains are VIFDs. We study the ring-theoretic properties of VIFDs and the $*$-operation analogs of VIFDs. Among them, we show that if $D$ is treed (resp., $*$-treed), then $D$ is a VIFD (resp., $*$-VIFD) if and only if $D$ is an ${\rm h}$-local Pr\"ufer domain (resp., a $*$-${\rm h}$-local P$*$MD) if and only if every nonzero prime ideal of $D$ contains an invertible (resp., a $*$-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.