Consider a noetherian ring $R$ and a collection $m_i$, $i\in I$ of maximal ideals of $R$. Let $P$ be a prime ideal of $R$. It is well-known that if the collection is finite (i.e. the index set $I$ is finite), then $P\subseteq \cup_{i\in I}m_i$ if and only if $P$ is contained in one of the $m_i$. (This is true under weaker assumptions.)

Now my question is what about the case of an infinite collection. Does the same hold (under suitable assumptions, e.g. $R$ is normal integral, etc.)? Or can anyone give a counter-example?