Lemma 1

Tag 050S, permanent. Cite it as [Bourbaki, Tag 050S].

Commutative Algebra, chapter VII, § 2. Dedekind domains

Let $A$ be a Noetherian domain in which every non-zeroprime ideal is maximal and $M$ afinitely generated torsion $A$ -module. Then the length $\operatorname{long}_A(M)$ of $M$ is infinite.

As $M$ is a torsion module, every prime ideal associated with $M$ is $\neq (0)$ and therefore maximal. The lemma then follows from Chapter IV, § 2, no. 5, Proposition 7.

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

Cited by

Lemma 3, § 2

Cites

Chapter IV, § 2, no. 5, Proposition 7