A finitely generated ideal of an integral domain

Question

Let $D$ be an integral domain and $Q(D)$ be the ring of fractions of $D$. If $I$ is an ideal of $D$ with the property that if for all ideals $J\not=0$ of $D$ such that $J\subseteq I$, there exists a subset $A_J$ of $Q(D)$ such that $A_JJ=I$. My professor said that $I$ must be a finitely generated ideal of $D$, but I could not prove that. Why is $I$ finitely generated ideal of $D$?

Answer 1

As you state this, you could have $J=I$, $A_J=\{1\}$ and $I$ a non-finitely generated ideal. So, the conclusion doesn't follow.

What is the case is that if $AI=D$ for some fractional ideal $A$, then $I$ is finitely generated.