Let $R$ be a commutative unitary ring and $M\_{I}$ be the intesection of all maximal ideals contains $I$.
Question: When for any two ideals $I$ and $J$ of $R$ there exists an ideal $K$ of $R$ such that $M\_{I}+M\_{J}=M\_{K}$?



Intersection of maximal ideals contains an ideal
A typical ring of dimension $2$, like $k[x,y]$, does not have that property. Indeed, if $I=(xy)$, $J=(x^2-y^2)$, then $M\_I=I$, $M\_J=J$, and $M\_I+M\_J=I+J$ contains $x^3$ and $y^3$ but not $x$ and $y$, so is not radical, so cannot be the intersection of any set of maximal ideals.


On the other hand, in a Dedekind domain, ideals of this form are exactly products of finitely many maximal ideals or the zero ideal. Scheme-theoretically, they correspond to finite sets of points on a curve or else the whole curve. This set is closed under addition, or, scheme-theoretically, intersection, so you're OK there.


Obviously, this also works in every local ring, where there is a unique maximal ideal.


