On $v$-domains: a survey

TL;DR

This survey comprehensively reviews the theory of $v$-domains, a class of integral domains that generalize Pr"{u}fer and Krull domains, including their characterizations, relationships, and historical development.

Contribution

It consolidates existing knowledge on $v$-domains, providing new characterizations and clarifying their connections with related classes of domains.

Findings

Multiple characterizations of $v$-domains are presented.

Relationships between $v$-domains and other domain classes are clarified.

Historical context and examples illustrate the development of the theory.

Abstract

An integral domain $D$ is a $v$--domain if, for every finitely generated nonzero (fractional) ideal $F$ of $D$, we have $(FF^{-1})^{-1}=D$. The $v$--domains generalize Pr\"{u}fer and Krull domains and have appeared in the literature with different names. This paper is the result of an effort to put together information on this useful class of integral domains. In this survey, we present old, recent and new characterizations of $v$--domains along with some historical remarks. We also discuss the relationship of $v$--domains with their various specializations and generalizations, giving suitable examples.