Let $R$ be a Noetherian domain, and let $I$ and $J$ be two ideals of $R$ such that their product $I\cdot J$ is a non-zero principal ideal. Is it true that $I$ and $J$ are principal ideals ? This seems an easy question to settle, but I can't find an answer.
Any idea is welcome, thanks !
Thanks a lot for your enlightening answers. I admit I'm more interested in a geometric setting (i.e. when $R$ is an algebra, finitely generated, or a localization of that). I fail to adapt examples coming from number theory to this setting. What do you think ?