A note on perinormal domains

TL;DR

This paper explores the properties of perinormal domains, establishing their relation to Pr"ufer v-multiplication domains and providing constructions to generate such domains, thereby advancing understanding of their structure.

Contribution

It proves that Pr"ufer v-multiplication domains are perinormal with no proper lying over overrings and characterizes treed perinormal domains as Pr"ufer domains, introducing new construction methods.

Findings

Pr"ufer v-multiplication domains are perinormal and have no proper lying over overrings.

A treed perinormal domain is a Pr"ufer domain.

Two pull-back constructions produce perinormal and non-perinormal domains.

Abstract

Recently, N. Epstein and J. Shapiro introduced and studied the perinormal domains: those domains A whose going down overrings are flat A-modules. We show that every Pr\"ufer v-multiplication domain is perinormal and has no proper lying over overrings. We also show that a treed perinormal domain is a Pr\"ufer domain. We give two pull-back constructions that produce perinormal/non-perinormal domains.