Consider the ring $R=\mathbb C[X,Y]$. For every two ideals $I,J$ of $R$, define $I*J:=\{ij : i\in I, j\in J\}$.
Now definitely, $I*J=J*I$ always holds. If $I$ is principal, then actually $I*J$ is an ideal of $R$.
My question is:
If $I$ is a proper, non-zero ideal of $R=\mathbb C[X,Y]$ such that for every ideal $J$ of $R$, $I*J$ is also an ideal of $R$ (i.e. $I*J=IJ$), then does it imply that $I$ is principal ? Or at least $I$ is contained in a principal prime ideal ? If neither of these are true, then can we characterize all ideals $I$ with the said property in some other way ?
Some thoughts towards possibly showing $I$ is principal : To show $I$ is principal, enough to show $I$ is free, then by Quillen-Suslin, enough to show $I$ is projective, and since we are in Noetherian, finitely generated case, enough to show $I$ is flat over $R$. So it is enough to show $I \otimes_R J \cong IJ =I*J$ for every ideal $J$. No idea how to show that though ...
