Does there exist a commutative ring with unity such that the pairwise intersection of distinct maximal ideals is the Jacobson radical?
Ie. if $M_1, M_2$ are any pair of distinct maximal ideals then $M_1 \cap M_2 = J(R)$.
And if that is false, is there a ring, $R$ and $a$ such that any collection of maximal ideals $|M| \geq a$ satisfies $\bigcap M = J(R)$.
Or more weakly is there a subset of maximal ideals such that the above is true even if it is not true for every maximal ideal.
I know the case where $a$ is finite fails when $R$ is semiprimitive and is trivially true for rings with at most $a$ maximal ideals but I am unable to find an answer otherwise.