Flat modules over valuation rings

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.