What are the maximal ideals containing the intersection of maximal ideals?

Question

Let $A$ be a finitely generated $k$-algebra where $k$ is a field. Let $M, M_i \in \operatorname{Specm}(A)$ be maximal ideals where $I$ is some index set.

Does $M \supseteq \bigcap_{i \in I} M_i$ imply $M = M_i$ for some $i \in I$?

The finite case is clear.

Answer 1

Let $A=ℂ[x]$ and let $M_i$ the set of all maximal ideals of the form $(x−a), a\neq 0$. Then $\cap M_i=0$, so $\cap M_i⊂(x)$, but $(x)\neq M_i$ for any $i$.