Exercise 8, § 2

Let $\mathbf{A}$ be an integral domain in which every prime ideal $\neq 0$ is invertible. Show that $\mathbf{A}$ is a Dedekind domain. (Prove first that every prime ideal $q' \neq 0$ of $\mathbf{A}$ is maximal, noting that, if $p$ is a prime ideal such that $q \neq p'$ and $p \supset p'$, then $p': p = p'$ (Chapter 11, § 1, Exercises 8 (b)) and on the other hand that $p': p = p'p^{-1}$, whence a contradiction is deduced. Deduce that $\mathbf{A}$ is Noetherian (Chapter II, § 1, Exercise 6 (b)) and finally apply Chapter II, § 5, no. 6, Theorem 4 to show that $\mathbf{A}_m$ is a field or a discrete valuation ring for every maximal ideal $m$ of $\mathbf{A}$.)