Research ArticleNo Access

Semirigid GCD domains II

Department of Mathematics, Idaho State University, Pocatello, 83209 ID, USA

https://doi.org/10.1142/S0219498822501614Cited by:1 (Source: Crossref)

Abstract

Let $D$ be an integral domain with quotient field $K,$ throughout $.$ Call two elements $x, y \in D ∖ {0}$ $v$ -coprime if $x D \cap y D = x y D .$ 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 $* :$ product of every pair of non- $v$ -coprime rigid elements is again rigid. Next, call $a \in D$ a valuation element if $a V \cap D = a D$ 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 $\sqrt{r D}$ 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 a variant of an SPD is a semirigid GCD domain.

Communicated by A. Facchini

Dedicated to the memory of Paul Cohn

Keywords:

AMSC: 13A05, 13F15, 13G05

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