Suppose we have a commutative ring $R$ with unit. I'm curious about what condition(s) on $R$ would be sufficient (without Axiom of Choice) to give a converse to the following familiar result:
(#) If $I,J$ are comaximal ideals of $R$ (i.e.: $I+J=R$), then $IJ=I\cap J$.
I discovered that the converse of (#) holds if $R$ is a PID, so that's sufficient, but I was wondering if that can be weakened at all, perhaps to Dedekind domain, UFD, or even integral domain. I suspect it doesn't hold in general.
In particular, if the converse of (#) holds when $R$ is a Dedekind domain (every non-unit ideal of $R$ can be uniquely factored as a product of prime ideals), can anyone give me a (sketch of a) proof?