Semirigid GCD domains II

Abstract

Let $D$ be an integral domain with quotient field $K,$ throughout$.$ Call two elements $x,y\in D\backslash \{0\}$ $v$-coprime if $xD\cap yD=xyD.$ Call a nonzero non unit $r$ of an integral domain $D$ rigid if for all $x,y|r$ we have $x|y$ or $y|x.$ Also call $D$ semirigid if every nonzero non unit of $D$ is expressible as a finite product of rigid elements. We show that a semirigid domain $D$ is a GCD domain if and only if $D$ satisfies $\ast :$ product of every pair of non-$v$-coprime rigid elements is again rigid. Next call $a\in D$ a valuation element if $aV\cap D=aD$ for some valuation ring $% V $ with $D\subseteq V\subseteq K$ and call $D$ a VFD if every nonzero non unit of $D$ is a finite product of valuation elements. It turns out that a valuation element is what we call a packed element: a rigid element $r$ all of whose powers are rigid and $\sqrt{rD}$ is a prime ideal. Calling $D$ a semi packed domain (SPD) if every nonzero non unit of $D$ is a finite product of packed elements, we study SPDs and explore situations in which an SPD is a semirigid GCD domain.