Exercise 3, § 7

Let $A_i \ (1 \leq i \leq n)$ be pairwise independent valuation rings of a field $K$ and let $A = \bigcap_i A_i$. For all $i$, let $a_i$ be an ideal $\neq 0$ of $A$, and write $a = \bigcap_i a_i;$

show that, for all $i, a_i = A_i a$. Conversely, if $b$ is an ideal $\neq 0$ of $A$ and $b_i = A_i b$, then $b = \bigcap_i b_i$ (use Theorem 1 of no. 2).