Example 3

Tag 000G, permanent. Cite it as [Bourbaki, Tag 000G].

Algebra, VIII, p. 5, § 1. Artinian Modules and Noetherian Modules

A principal ideal domain (VII, §1, No. 1, p. 1, Definition 1) is Noetherian. An integral domain A that is not a field is not an Artinian ring: for every nonzero noninvertible element $a$ of A, the sequence of ideals $a^nA$ (for $n\in \mathbf{N}$ ) is strictly decreasing. In particular, the ring $\mathbf{Z}$ of integers is Noetherian but not Artinian.

Bourbaki sets the proof directly after the statement, and the corpus does not separate them, so what is above is the statement and its proof together.

Cited by

Example 3, § 1
Corollary, § 1

Cites

Nothing.