Flat modules over valuation rings

Let RR be a valuation ring and let QQ be its total quotient ring. It is proved that any singly projective (respectively flat) module is finitely projective if and only if QQ is maximal (respectively artinian). It is shown that each singly projective module is a content module if and only if any non-unit of RR is a zero-divisor and that each singly projective module is locally projective if and only if RR is self-injective. Moreover, RR 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 ππ-coherent.A complete characterization of semihereditary commutative rings which are ππ-coherent is given. When RR is a commutative ring with a self-FPFP-injective quotient ring QQ, it is proved that each flat RR-module is finitely projective if and only if QQ is perfect.