Exercise 6, § 1

Let $\mathbf{A}$ be an integral domain.

(a) For a divisorial ideal $a$ of $\mathbf{A}$ to be such that $\mathrm{div}(a)$ is invertible in $D(A)$, it is necessary and sufficient that $a: a = \mathbf{A}$ (cf. Exercise 5 and Algebra, Chapter VI, § 1, Exercise 30). In particular, in order that $a: a = \mathbf{A}$ for every divisorial ideal $a$ of $\mathbf{A}$, it is necessary and sufficient that $\mathbf{A}$ be completely integrally closed.

(b) In order that $a: a = \mathbf{A}$ for every finitely generated ideal $a \neq 0$ of $\mathbf{A}$, it is necessary and sufficient that $\mathbf{A}$ be integrally closed.