Exercise 8, § 4

Let A be an integral domain, K its field of fractions. Let B be a ring and $\varphi : A \to B$ a ring homomorphism making B into a faithfully flat A-module. Suppose that B has only a finite number of minimal prime ideals $p_1, ..., p_n$. For $1 \leq i \leq n$, put $\overline{B}_i = B/p_i$, let $L_i$ denote the field of fractions of $B_i$ and L the product ring of the $L_i$. Finally, let $\overline{A}$ denote the integral closure of A, $\overline{B}_i$ that of $B_i$. a) The homomorphism from A into L induced by $\varphi$ is injective, and extends to an injective homomorphism $\psi$ from K into L. b) We have $\overline{A} = \psi^{-1}(\prod_{i=1}^n \overline{B}_i)$. (One may prove that if $x \in \psi^{-1}(\prod_{i=1}^n \overline{B}_i)$ and if $x'$ is its image in $K \otimes_A B$, there exists a monic polynomial P with coefficients in B such that $P(x')$ is nilpotent in $K \otimes_A B$.)