Proposition 7

Let $A$ be a Krull domain, $(v_P)_{P \in P(A)}$ the family of its essential valuations and $a \in I(A)$. Then the coefficient of $P$ in $\operatorname{div} a$ is $\inf_{y \in a} (v_P(y))$. If $p$ is the divisorial prime ideal corresponding to the extremal divisor $P$, then $aA_p = \tilde{a}A_p$.

As $a = \sum_{y \in a} Ax$, Proposition 2 (b) (no. 1) shows that $\operatorname{div}(a) = \inf_{x \in a} (\operatorname{div}(Ax))$, whence our first assertion. The second follows immediately, since $\operatorname{div} \tilde{a} = \operatorname{div} a$ and $A_p$ is the ring of the discrete valuation $v_P$.