Exercise 4, § 1

Let $\mathbf{A}$ be a completely integrally closed domain and $m$ a maximal ideal of $\mathbf{A}$. Show that, either $m$ is invertible, or $m$ is not divisorial and $\mathbf{A}: m = \mathbf{A}$. (Note that, if $m$ is not invertible, necessarily $rn. (A:m)^k = m$ for every integer $k > 0$.)