Let $R$ be a factorization domain in which every ideal generated by two elements is a principal domain. Show that $R$ is a PID.
So let $I \subset R$ be an Ideal, then I have to show that there exists $a \in R$ so that $I=(a)$. I need some hint. What do I know about an arbitray Ideal $I$, so that I can use the premise and show the existence of such an $a$?