Proposition 2

Let $A$ be a noetherian ring, $m$ a positive integer and $a$ an ideal contained in the radical of $A$ and generated by $m$ elements. One has

The inequality $\dim(A/a) \leq \dim(A)$ follows from prop. 6 of § 1, no 3. An immediate induction on $m$ shows that it is enough to establish the inequality

for every element $x$ of the radical of $A$, that is to say, to prove that one has $\dim(A/xA) \geq n - 1$ for every chain $p_0 \subset ... \subset p_n$ of prime ideals of $A$, of length $n \geqslant 1$, and such that $x \in p_n$. It suffices to construct a chain $q_1 \subset ... \subset q_n$ of prime ideals of $A$, with $x \in q_1$ and this follows from the following lemma: