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.
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Infinite CRT fails