Proposition 3

Let $h : A \to A'$ be a ring homomorphism such that $A'$ is a finitely generated $A$-module; then, for every prime ideal $\mathfrak{p}$ of $A$, the set of prime ideals of $A'$ lying above $\mathfrak{p}$ is finite.

Let $S = A - p$; by Lemma 1 we may replace $A$ by $S^{-1}A$, $A'$ by $S^{-1}A'$ (which is a finitely generated $S^{-1}A$-module) and $p$ by $S^{-1}p$; in other words, we may assume that $A$ is a local ring and $p$ is its maximal ideal. Then (by the remark made at the beginning of this no.) $A$ may be replaced by $A/p$, $A'$ by $A'/pA'$ and $p$ by $(0)$, for $A'/pA' = (A/p) \otimes_A A'$ is a finitely generated $(A/p)$-module. Thus we have finally reduced the problem to proving the proposition when $A$ is a field and $p = (0)$; $A'$ is then an $A$-algebra of finite rank and therefore Artinian and we know that in such an algebra there is only a finite number of prime ideals (Chapter IV, § 2, no. 5, Proposition 9).