Exercise 18, § 2

Let $A$ be a ring, the union of a right directed family of subrings $A_i$, which are Dedekind domains. Suppose that, for every prime ideal $p$ of $A_i$, there exists $\beta \geq a$ such that there are at least two distinct prime ideals of $A$, lying above $p$.

(a) Show that the ring of all algebraic integers (the integral closure of $\mathbf{Z}$ in $C$) satisfies the above conditions (cf. Chapter V, $§ 2$, Exercise 6).

(b) Let $v$ be a non-improper valuation on $A$, $z \in A$ such that $v(z) > 0$ and let $a$ be the ideal of $A$ consisting of the $x$ satisfying $v(x) \geq v(z)$. Show that $A : a = A$ and hence that $a$ is not invertible, although for every maximal ideal $m$ of $A$, $aA_m$ is a principal ideal in the valuation ring $A_m$ (cf. Chapter 11, $§ 5$, no. 6, Theorem 4).

(c) Deduce from (a) and (b) an example of a Priiferian domain which is completely integrally closed but not a Krull domain (cf. Exercises 15 (c) and 17 (b) and Chapter V, $§ 1$, Exercise 14).