Let $A$ be a Noetherian normal ring, that is, the localization of $A$ at every prime is a normal domain. I want to show $A$ is a finite product of normal domains.
If $p_1,\ldots,p_n$ are the minimal primes of $A$, I can show $$A \cong A/p_1 \times \cdots \times A/p_n.$$ However why should each $A/p_i$ be a normal domain? I can't seem to see why this is true. I need to see why given any prime $p \supseteq p_i$, $(A/p_i)_p = A_p /{p_i}_p$ is normal. Why should a quotient of a normal domain be normal?
You know that $A\simeq A_1\times\cdots\times A_n$ with $A_i$ integral domains. Let $m_i$ be a maximal ideal of $A_i$. Then $M_i=A_1\times\cdots\times m_i\times\cdots\times A_n$ is a maximal ideal of $A$ and $A_{M_i}\simeq A_{i_{m_i}}$, so $A_{i_{m_i}}$ is a normal domain. Can you prove now that $A_i$ is normal?