Exercise 3, § 1

Let $A$ be a Noetherian integral domain and $p$ a prime ideal of height 1.

Show that $\mathbf{A}: p \neq \mathbf{A}$ (in the field of fractionsof $\mathbf{A}$). (Ifc $\neq 0$ is an element of $p$, note that $p \in \mathrm{Ass}(A/Ac)$ and deduce that $Ac: p \# Ac$ by virtue of Chapter IV, § 2, Exercise 30). Is the result still valid for a non-Noetherian integral domain (consider a non-discrete valuation ring of height 1)?