Example 1

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

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

One has $\dim(A) < 0$ if and only if $A$ is reduced to 0. In order that one have $\dim(A) \leqslant 0$ , it is necessary and sufficient that every prime ideal of $A$ be maximal. Integral rings of dimension 0 are fields. A noetherian ring is of dimension $\leqslant 0$ if and only if it is artinian (IV, § 2, No. 5, prop. 9).

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

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