Let $I,J$ two ideals in a ring $R$. The product of ideals $IJ$ is included in $I \cap J$. For example we have equality in $\mathbb{Z}$ if generators have no common nontrival factors, in a ring $R$ when $I+J=(1)$, or when $R/IJ$ has no nonzero nilpotent elements. My question is not about equality, instead it is about strict inclusion.
Under what conditions $IJ \subsetneq I \cap J$ ?
If the question appears a little too general, then my primary aim is to see what happens under the hypothesis that $R$ is a Dedekind domain.
