Let $R$ be a Dedekind domain. We know that for ideals $I$ and $J$ of $R$ we have $I\mid J \iff I \supseteq J$. This fact is used for example in Marcus' Number Fields to show that we have unique factorization of ideals in $R$. However its proof for me is element based and doesn't seem very slick.
My question is:
Can we show $I\mid J \iff I \supseteq J$ using localization (not knowing about unique factorization of ideals in $R$)?
If this is possible, I would only like hints on how to do it as I would like to complete the problem myself.
Set $K=J:I$. Then $K_{\mathfrak p}=J_{\mathfrak p}:I_{\mathfrak p}$. Since $R_{\mathfrak p}$ is a DVR we get $I_{\mathfrak p}K_{\mathfrak p}=J_{\mathfrak p}$ and we are done.