Exercise 3, § 2

Let $A$ be a Dedekind domain, $K$ its field of fractions, $P$ the set of prime ideals $\neq 0$ of $A$ and for all $\varrho \in P$ let $v_\varrho$ be the corresponding essential valuation on $K$. For every sub-A-module $M$ of $K$ and all $\varrho \in P$ we write $v_\varrho(M) = \inf v_\varrho(x)$ (taken in $\overline{\mathbf{R}}$); if $M \neq 0$, $v_\varrho(M) < +\infty$ for all $\varrho \in P$ and $v_\varrho(M) \leq 0$ for almost all $\varrho \in P$. If $M, N$ are two sub-A-modules of $K$, show that the relation $M \subset N$ is equivalent to $v_\varrho(N) \leq v_\varrho(M)$ for all $\varrho \in P$ (use Proposition 9 of § 1, no. 5). Conversely, for every family $(v_\varrho)_{\varrho \in P}$ of elements equal to an integer or $-\infty$ and such that $v_\varrho \leq 0$ for almost all $\varrho \in P$, there exists a unique sub-A-module $M$ of $K$ such that $v_\varrho(M) = v_\varrho$ for all $\varrho \in P$.