Corollary 3

Let $h : A \to A'$ be a ring homomorphism such that $A'$ is integral over $A$ and $m$ a maximal ideal of $A$ and suppose that there are in $A'$ only a finite number of distinct maximal ideals $m'_j$ ($1 \leq j \leq n$) lying above $m$. Let $q'_j$ be the saturation of $mA'$ with respect to $m'_j$ (Chapter II, § 2, no. 4). Then: (i) In the ring $A'/q'_j$ the divisors of zero are the elements of $m'_j/q'_j$ and they are nilpotent ($1 \leq j \leq n$). (ii) $mA' = \bigcap_j q'_j = \prod_j q'_j$. (iii) The canonical homomorphism $A'/mA' \to \prod_j (A'/q'_j)$ is bijective.

For a prime ideal of $A'$ to contain $mA'$, it is necessary and sufficient that its inverse image under $h$ contain $m$ and hence that it lie above $m$, since $m$ is maximal in $A$; the $m'_j$ are therefore the only prime ideals of $A'$ containing $mA'$ (Proposition 1) and therefore $c' = \bigcap_j m'_j$ is the radical of $mA'$ (Chapter II, § 2, no. 6, Corollary 1 to Proposition 13). By definition of $q'_j$, the class mod. $q'_j$ of an element of $A' - m'_j$ is not a divisor of 0 in $A'/q'_j$; on the other hand, as the $m'_j$ are distinct maximal ideals, for every index $j$ there exists an element $a'_j$ belonging to $\bigcap_j m'_j$ and not to $m'_j$ (Chapter II, § 1, no. 1, Proposition 4): then, for all $x \notin m'_j, a'_j x \in r'$, hence the class mod. $q'_j$ of $a'_j x$ is nilpotent and, as that of $a'_j$ is not a divisor of 0, we conclude that the class of $x$ is nilpotent; in other words $m'_j$ is the radical of $q'_j$, which proves (i). It follows that the $q'_j$ are relatively prime in pairs (Chapter II, § 1, no. 1, Proposition 3); (iii) will therefore be a consequence of (ii), taking account of Chapter II, § 1, no. 2, Proposition 5. To establish (ii), we note that in the ring $A'/mA'$ the $m'_j/mA'$ are the only maximal ideals and $q'_j/mA'$ is the saturation of (0) with respect to $m'_j/mA'$ (Chapter 11, § 2, no. 4); we may therefore restrict our attention to the case $mA' = (0)$; the assertion of (ii) then follows from Chapter II, § 3, no. 3, Corollary 2 to Theorem 1.