Take the tour ×  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 ?
share|improve this question Dear Lierre: You may want to assume
that your ideals are nonzero. –  Pierre-Yves Gaillard Aug 1 '12
at 9:33 Thanks, I wrote proper meaning non-zero. –  Lierre Aug 1
'12 at 9:41 $(x,y)z=0$ in $\mathbb Z[X,Y,Z]/(XZ,YZ)$ (obvious
notation). –  Pierre-Yves Gaillard Aug 1 '12 at 9:59 Thanks, but
the product ideal is zero. –  Lierre Aug 1 '12 at 10:06 In an
abelian semi-group a product of elements is invertible if and
only if all