Theorem 1

Let $h : A \to A'$ be an injective ring homomorphism such that $A$ is integral over $A$ and $p$ a prime ideal of $A$. There exists a prime ideal $p'$ of $A'$ lying above $p$.

Suppose first that $A$ is a local ring and $p$ the maximal ideal of $A$; then, for every maximal ideal $m'$ of $A'$, $h^{-1}(m')$ is a maximal ideal of $A$ (Proposition 1) and hence equal to $p$, which proves the theorem in this case (since $A'$ contains $A$ by hypothesis and is therefore not reduced to 0). In the general case, let us write $S = A - p$; then $S^{-1}A$ is a local ring whose maximal ideal is $S^{-1}p$ (Chapter 11, § 2, no. 5, Proposition 11), $S^{-1}h : S^{-1}A \to S^{-1}A$, is injective (Chapter II, § 2, no. 4, Theorem 1) and $S^{-1}A'$ is integral over $S^{-1}A$ ($§ 1$, no. 5, Proposition 16); then there exists a prime ideal $q'$ of $S^{-1}A'$ lying above $S^{-1}p$ and we know that $q' = S^{-1}p'$, where $p'$ is a prime ideal of $A'$ lying above $p$ (Lemma 1).

If $h : A \to A'$ is not injective, Theorem 1 is no longer necessarily true, as the example of the homomorphism $Z \to \mathbf{Z}/n\mathbf{Z} \ (n > 1)$ shows. However Theorem 1 can be applied to the canonical injection $h(A) \to A'$; in other words, the statement of Theorem 1 is true for prime ideals $p$ containing $\mathrm{Ker}(h)$.