Proposition 5

Let $A$ be a Krull domain, $K$ its field of fractions and $(v_P)_{P \in P(A)}$ the family of its essential valuations. Let $(n_P)_{P \in P(A)}$ be a family of rational integers which are zero except for a finite number of indices. Then the set of $x \in K$ such that $v_P(x) \geq n_P$ for all $P \in P(A)$ is the divisorial ideal $a$ of $A$ such that $\text{div } a = \sum_{P \in P(A)} n_P \cdot P$.

Let $x \in K^*$. In order that $x \in a$, it is necessary and sufficient that $Ax \subset a$, hence that $\text{div}(x) \geq \text{div } a$ and hence, by (4), that $v_P(x) \geq n_P$ for all $P \in P(A)$.