Exercise 7, § 2

Let $\mathbf{A}$ be a Noetherian domain. Show that the following properties are equivalent: (α) $\mathbf{A}$ is a Dedekind domain. (β) For every maximal ideal $m$ of $\mathbf{A}$, there exists no ideal $a$ distinct from $m$ and $m^2$ such that $m^2 \subset a \subset m$. (γ) For every maximal ideal $m$ of $\mathbf{A}$, the set of primary ideals for $m$ is totally ordered by inclusion. (δ) For every maximal ideal $m$ of $\mathbf{A}$, every primary ideal for $m$ is a product of prime ideals. (Prove that each of the properties (γ) and (δ) implies (β); then show that (β) implies that $\mathbf{A}_{m}$ is a field or a discrete valuation ring (cf. Chapter VI, § 3, no. 5, Proposition 9).) Give an example of a local integral domain satisfying properties (β), (γ) and (δ), which is not a valuation ring (cf. Chapter VI, § 3, Exercise 7).