Completely integrally closed Prufer $v$-multiplication domains

TL;DR

This paper investigates the conditions under which power series rings over PVMDs are integrally closed, providing characterizations of completely integrally closed PVMDs and their relation to other domain classes.

Contribution

It establishes new equivalences and characterizations for completely integrally closed PVMDs, especially in relation to power series rings and torsion $t$-class groups.

Findings

A PVMD is completely integrally closed iff the intersection of all powers of a proper $t$-invertible $t$-ideal's $v$-closure is zero.

For an AGCD domain, $D[[X]]$ is integrally closed iff $D$ is a completely integrally closed PVMD with torsion $t$-class group.

Certain PVMD classes are characterized by the equivalence of being Archimedean and being completely integrally closed.

Abstract

We study the effects on $D$ of assuming that the power series ring $D[[X]]$ is a $v$-domain or a PVMD. We show that a PVMD $D$ is completely integrally closed if and only if $\bigcap_{n=1}^{\infty }(I^{n})_{v}=(0)$ for every proper $t$-invertible $t$-ideal $I$ of $D$. Using this, we show that if $D$ is an AGCD domain, then $D[[X]]$ is integrally closed if and only if $D$ is a completely integrally closed PVMD with torsion $t$-class group. We also determine several classes of PVMDs for which being Archimedean is equivalent to being completely integrally closed and give some new characterizations of integral domains related to Krull domains.