Semirigid GCD domains II

TL;DR

This paper characterizes semirigid GCD domains through a specific product property of rigid elements and explores the structure of semi packed domains, connecting valuation elements, packed elements, and domain classifications.

Contribution

It introduces the concept of semi packed domains and provides a characterization of semirigid GCD domains via a product condition on rigid elements.

Findings

Semirigid domains are GCD domains if and only if a specific product condition holds.

Valuation elements are characterized as packed elements with particular properties.

The paper explores conditions under which semi packed domains are semirigid GCD domains.

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.