Semirigid GCD domains II

Abstract

Let [Formula: see text] be an integral domain with quotient field [Formula: see text] throughout[Formula: see text] Call two elements [Formula: see text][Formula: see text]-coprime if [Formula: see text] Call a nonzero non-unit [Formula: see text] of an integral domain [Formula: see text] rigid if for all [Formula: see text] we have [Formula: see text] or [Formula: see text] Also, call [Formula: see text] semirigid if every nonzero non-unit of [Formula: see text] is expressible as a finite product of rigid elements. We show that a semirigid domain [Formula: see text] is a GCD domain if and only if [Formula: see text] satisfies [Formula: see text] product of every pair of non-[Formula: see text]-coprime rigid elements is again rigid. Next, call [Formula: see text] a valuation element if [Formula: see text] for some valuation ring [Formula: see text] with [Formula: see text] and call [Formula: see text] a VFD if every nonzero non-unit of [Formula: see text] is a finite product of valuation elements. It turns out that a valuation element is what we call a packed element: a rigid element [Formula: see text] all of whose powers are rigid and [Formula: see text] is a prime ideal. Calling [Formula: see text] a semi-packed domain (SPD) if every nonzero non-unit of [Formula: see text] is a finite product of packed elements, we study SPDs and explore situations in which a variant of an SPD is a semirigid GCD domain.