# Flat modules over valuation rings - Mathematics > Rings and Algebras

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Abstract: Let $R$ be a valuation ring and let $Q$ be its total quotient ring. It isproved that any singly projective respectively flat module is finitelyprojective if and only if $Q$ is maximal respectively artinian. It is shownthat each singly projective module is a content module if and only if anynon-unit of $R$ is a zero-divisor and that each singly projective module islocally projective if and only if $R$ is self injective. Moreover, $R$ ismaximal if and only if each singly projective module is separable, if and onlyif any flat content module is locally projective. Necessary and sufficientconditions are given for a valuation ring with non-zero zero-divisors to bestrongly coherent or $\pi$-coherent. A complete characterization ofsemihereditary commutative rings which are $\pi$-coherent is given. When $R$ isa commutative ring with a self FP-injective quotient ring $Q$, it is provedthat each flat $R$-module is finitely projective if and only if $Q$ is perfect.

Autor: Francois Couchot LMNO

Fuente: https://arxiv.org/