# Infinite CRT fails

Construct a commutative ring $A$ and a countable infinite family of distinct maximal ideals $\mathfrak m_i,i=1,2,3,\ldots$ of $A$ such that

1.  The intersection $\bigcap \mathfrak m_i$ is zero, yet
2.  The canonical projection $A\to \prod A/\mathfrak m_i$ is not surjective.