Does $I(J\cap K)=IJ\cap IK$ hold in a Dedekind ring?

Question

For ideals in any ring, we have the relation $I(J\cap K)\subseteq IJ\cap IK$. Do we actually have equality if we are in a Dedekind domain?

I've been looking around for a reference, but haven't found anything. I haven't been able to construct a counterexample (if there is one).

Answer 1

Hint:

$$\mathfrak{p}_1^{e_1}\cdots\mathfrak{p}_m^{e_m}\cap\mathfrak{p}_1^{f_1}\cdots\mathfrak{p}_m^{f_m}=\mathfrak{p}_1^{\max\{e_1,f_1\}}\cdots\mathfrak{p}_m^{\max\{e_m,f_m\}}$$

and

$$a+\max\{b,c\}=\max\{a+b,a+c\}$$

if $a,b,c\geqslant 0$.