On super $v$-domains

TL;DR

This paper characterizes super $v$-domains as those domains where all quotient rings are $v$-domains, and establishes their equivalence with polynomial and certain extension domains, providing new examples between $v$-domains and P-domains.

Contribution

It introduces the concept of super $v$-domains, characterizes them as locally $v$-domains, and links their properties to polynomial and extension domains, with new examples.

Findings

Super $v$-domains are characterized as locally $v$-domains.

Super $v$-domains are equivalent to their polynomial and extension domains being super $v$-domains.

New examples of super $v$-domains are provided, lying between $v$-domains and P-domains.

Abstract

An integral domain $D,$ with quotient field $K,$ is a $v$-domain if for each nonzero finitely generated ideal $A$ of $D$ we have $(AA^{-1})^{-1}=D.$ It is well known that if $D$ is a $v$-domain$,$ then some quotient ring $D_{S}$ of $D$ may not be a $v$-domain. Calling $D$ a super $v$-domain if every quotient ring of $D$ is a $v$-domain we characterize super $v$-domains as locally $v$-domains. Using techniques from factorization theory we show that $D$ is a super $v$-domain if and only if $D[X]$ is a super $v$-domain if and only if $D+XK[X]$ is a super $v$-domain and give new examples of super $v$ -domains that are strictly between $v$-domains and P-domains that were studied in [Manuscripta Math. 35(1981)1-26]