# Is there a ring in which the pairwise intersection of maximal ideals is the Jacobson radical

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.

• Yes, that' what pairwise means. – user661541 Aug 31 '19 at 17:36

If there exists a set of $$n$$ distinct maximal ideals intersecting to $$J(R)$$, then the Chinese remainder theorem says that $$R/J(R)$$ is isomorphic to $$n$$ fields, and such a ring has exactly $$n$$ maximal ideals.
So if just one set of $$n$$ exists, they are precisely the entire set of maximal ideals.