A family of maximal ideals

Let $m_i$, $i \in I,$ be an infinite family of maximal ideals in a commutative ring with identity (it is not supposed to be Noetherian). When does there exist $j \in I$ such that $\cap_{i\not= j} m_i\subseteq m_j$? Or is there any equivalent condition for this?

• A reformulation. You have an infinite subset $Z$ of $X = Spec(A)$ with the Zariski topology, consisting of closed points. You want there to be an element $z \in Z$ which lies in the Zariski closure of $Z - \{z\}$. – user91132 Nov 27 '15 at 19:58