Exercise 25, § 1

An integrally closed domain $A$ is said to be of finite character if there exists a family $(v_i)_{i \in I}$ of valuations on the field of fractions of $A$ satisfying properties (AK$_{\text{II}}$) and (AK$_{\text{III}}$).

(a) Show that, if $A$ is integrally closed and of finite character, for every element $x \neq 0$ in $A$, there can only be a finite number of extremal elements of the ordered group of principal divisors which are $\leq \operatorname{div}(x)$. (Let $J \subset I$ be the finite set of indices such that $v_i(x) > 0$. If $m$ is the number of elements in $J$, show that there cannot be $m + 1$ elements $y_j$ ($1 \leq j \leq m + 1$) dividing $x$ and such that the elements $\operatorname{div}(y_j)$ are extremal, observing that, for all $j$, $\operatorname{div}(y_j)$ must be prime to $\sum_{k \neq j} \operatorname{div}(y_k)$.)

(b) Show that the ring of integral functions of one complex variable (Chapter V, § 1, Exercise 12) is a pseudo-principal domain (Exercise 21) but not a domain of finite character (use (a)). \*