Flat modules over valuation rings

TL;DR

This paper investigates properties of flat and projective modules over valuation rings, establishing conditions under which these modules have specific finiteness, coherence, and projectivity properties, with implications for ring structure.

Contribution

It provides new characterizations of flat and singly projective modules over valuation rings, linking module properties to ring-theoretic conditions like maximality, artinianity, and coherence.

Findings

Singly projective modules are finitely projective iff the total quotient ring is maximal.

Each singly projective module is a content module iff non-units are zero-divisors.

Conditions for valuation rings to be strongly coherent or $oldsymbol{ extpi}$-coherent are established.

Abstract

Let $R$ be a valuation ring and let $Q$ be its total quotient ring. It is proved that any singly projective (respectively flat) module is finitely projective if and only if $Q$ is maximal (respectively artinian). It is shown that each singly projective module is a content module if and only if any non-unit of $R$ is a zero-divisor and that each singly projective module is locally projective if and only if $R$ is self injective. Moreover, $R$ is maximal if and only if each singly projective module is separable, if and only if any flat content module is locally projective. Necessary and sufficient conditions are given for a valuation ring with non-zero zero-divisors to be strongly coherent or $\pi$-coherent. A complete characterization of semihereditary commutative rings which are $\pi$-coherent is given. When $R$ is a commutative ring with a self FP-injective quotient ring $Q$, it is proved that each flat $R$-module is finitely projective if and only if $Q$ is perfect.