Remark 1

Tag 0CKH, permanent. Cite it as [Bourbaki, Tag 0CKH].

Commutative Algebra, chapter VIII, § 1. Dimension de Krull d’un anneau

Let A be a noetherian ring and M a finitely generated A-module. It is equivalent to say that $\dim_A(M) \leq 0$ , or that the elements of Supp(M) are maximal ideals of A, or that M is of finite length (IV, § 2, n° 5, prop. 7).

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

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