Proposition 11

Let $\mathbf{A}$ be a local Krull domain (in particular an integrally closed local Noetherian domain) and $m$ its maximal ideal. The following conditions are equivalent: (a) $\mathbf{A}$ is discrete valuation ring; (b) $m$ is invertible; (c) $\mathbf{A}:m \neq \mathbf{A}$; (d) $m$ is divisorial; (e) $m$ is the only non-zero prime ideal of $\mathbf{A}$.

As every non-zero ideal of a discrete valuation ring is principal (Chapter VI, § 3, no. 6, Proposition 9), it is invertible and hence (a) implies (b). If $m$ is invertible, its inverse is $\mathbf{A}:m$ (Chapter II, § 5, no. 6, Proposition 10) and hence $\mathbf{A}:m \neq \mathbf{A}$; hence (b) implies (c). If $\mathbf{A}:m \neq \mathbf{A}$, then $\mathbf{A}:(\mathbf{A}:m) \neq \mathbf{A}$; now $m \subset \mathbf{A}:(\mathbf{A}:m)$; hence $m = \mathbf{A}:(\mathbf{A}:m)$ since $m$ is maximal, so that $m$ is divisorial (no. 1, Proposition 1 (c)); thus (c) implies (d). The fact that (d) implies (e) follows from Theorem 3 of no. 6, Finally, if $m$ is the only non-zero prime ideal of $\mathbf{A}$, it is of height 1 and hence $\mathbf{A}_m$ is a discrete valuation ring (no. 6, Theorem 4); as $\mathbf{A}$ is local, $\mathbf{A}_m = \mathbf{A}$, which shows that (e) implies (a).