Corollary

Tag 04YY, permanent. Cite it as [Bourbaki, Tag 04YY].

Commutative Algebra, chapter VII, § 1. Krull domains

For a Noetherian ring to be a Krull domain, it is necessary and sufficient that it be an integrally closed domain.

An integrally closed Noetherian domain is completely integrally closed (Chapter V, § 1, no. 4).

There are non-Noetherian Krull domains, for example the polynomial ring $K[X_n]_{n \in \mathbf{N}}$ over a field $K$ in an infinity of indeterminates (cf. Exercise 8).

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

Cited by

§ 4. Modules over integrally closed Noetherian domains

Cites

Chapter V, § 1, no. 4
Exercise 8