Lemma 1

Let $A$ be a valuation ring. Every finitely generated torsion-free $A$-module is free. Every finitely generated ideal $\mathfrak{a}$ of $A$ is principal. Every torsion-free $A$-module is flat.

Let $E$ be a finitely generated torsion-free $A$-module and let $x_1, \ldots, x_n$ be generators of $E$ which are minimal in number; we show that they are linearly independent. If $\sum_{i=1}^n a_i x_i = 0$ ($a_i \in A$) is a non-trivial relation between the $x_i$, one of the $a_i$, say $a_1$, divides all the others since the set of principal ideals of $A$ is totally ordered by inclusion (§ 1, no. 2, Theorem 1); then $a_1 \neq 0$ since the relation is non-trivial. As $E$ is torsion-free, we can divide by $a_1$, which amounts to assuming that $a_1 = 1$. But then $x_1$ is a linear combination of $x_2, \ldots, x_n$, contrary to the minimal character of $n$. Hence $E$ is free.

In particular every finitely generated ideal $\mathfrak{a}$ of $A$ is principal, all the elements of a system of generators of $\mathfrak{a}$ being multiples of one of them. Proposition 3 of Chapter I, § 2, no. 4 then shows that every torsion-free $A$-module is flat.