Remark 1

Tag 0CK6, permanent. Cite it as [Bourbaki, Tag 0CK6].

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

Let $\rho : A \to B$ be a homomorphism of rings. Then dim(B) is the least upper bound of the numbers dim(B/ $\rho(p)$ .B), where p runs through the set of minimal prime ideals of A: in fact, for every minimal prime ideal q of B, there exists a minimal prime ideal p of A contained in $\rho^{-1}(q)$ (II, § 2, No. 6, Lemma 2), and one has

\dim(B/q) \leq \dim(B/\rho(p).B) \leq \dim(B)

by Proposition 6, a); one concludes by Proposition 6, c).

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

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