Exercise 12, § 4

Let A be an integral domain, $\overline{A}$ its integral closure. Let $a_1, ..., a_r$ be nonzero elements of $\overline{A}$, b the fractional ideal $\sum_{i=1}^r Aa_i$ of A. a) There exists a fractional ideal $a$ of A such that $aa_i \subset a$ for $1 \leq i \leq r$. One also has $\tilde{a}a_i \subset \tilde{a}$ for $1 \leq i \leq r$; we recall (VII, § 1, No. 1) that $\tilde{a}$ is the intersection of the principal fractional ideals containing $a$. b) Let $y$ be a nonzero element of $\tilde{b}$. Then

c) One has $\tilde{b} \subset \overline{A}$. d) Let $z$ be an element of $\overline{A} : \overline{A}b$. Then $\overline{A}$ contains $\tilde{b}z$. e) One has $b \subset \overline{A} : (\overline{A} : \overline{A}b)$. f) Let $\mathfrak{P}$ be a divisorial ideal of $\overline{A}$ and $p = \mathfrak{P} \cap A$. Then $p$ is divisorial in A. (By e), one obtains $\tilde{p} \subset \mathfrak{P}$. But one also has $\tilde{p} \subset A$.