Corollary

Under the hypotheses on $A$ and $A'$ of Theorem 3, let $p$ be a prime ideal of $A$. The prime ideals of $A'$ lying above $p$ are the minimal elements of the set $\mathscr{E}$ of prime ideals of $A'$ containing $pA'$.

A prime ideal of $A'$ lying above $p$ is minimal in $\mathscr{E}$ by virtue of no. 1, Corollary 1 to Proposition 1. Conversely, let $q'$ be a minimal element of $\mathscr{E}$. As $q' \cap A \supset p$, Theorem 3 shows that there exists a prime ideal $p'$ of $A'$ lying above $p$ such that $q' \supset p'$. As $p'$ contains $pA'$, the hypothesis made on $q'$ implies that $q' = p'$ and hence $q'$ lies above $p$.

\* Let $V, V'$ be two affine algebraic varieties and $f$ a morphism of $V'$ to $V$ such that $f(V')$ is dense in $V$. Let $A$ (resp. $A'$) be the ring of functions regular on $V$ (resp. $V'$); having been given $f$ we can identify $A$ with a sub-ring of $A'$; suppose that $A'$ is integral over $A$. Theorem 1 of no. 1 shows that for every irreducible subvariety W of V there exists an irreducible subvariety W' of V' such that f(W') is a dense subset of W; in particular, every point of V is the image of an irreducible subvariety of V', which shows that f is surjective. Similarly, the restriction of f to every irreducible subvariety W' of V' maps W' onto an irreducible subvariety of V. Corollary 2 to Theorem 1, no. 1 shows that, if W and X are two irreducible subvarieties of V such that W ⊃ X and if W' is an irreducible subvariety of V' such that f(W') = W, then there exists an irreducible subvariety X' of V' contained in W' and such that f(X') = X.

If A is integrally closed, V is called normal. Theorem 3 shows that, if V is normal, if W and X are irreducible subvarieties of V such that W ⊃ X and if X' is an irreducible subvariety of V' such that f(X') = X, then there exists a subvariety W' of V' containing X' and such that f(W') = W. Finally, the Corollary to Theorem 3 shows that, if V is normal and W is an irreducible subvariety of V, the irreducible subvarieties W' of V' such that f(W') = W are just the irreducible components of $\overline{f}(W)$.