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Theorem $11.2$ in Eisenbud states the following:

A Noetherian domain $R$ is normal iff for every prime $P$ of $R$ associated to a principal ideal, $P_P$ is principal.

Since $R$ is an integral domain, then for any principal ideal $Q=(r)$, the associated primes must all be zero since an integral domain has no zero divisors. Furthermore, localizing $0$ at $0$ is just zero...

What is going on here??

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    Did you try looking for "associated'' in the index to see what is meant? (Obviously the interpretation you give isn't the correct one.) I don't have a copy of the book, but looking at the discussion preceding the theorem on google books, I would guess that it means an associated prime of $R/I$ (for a principal ideal $I$). A little googling seemed to confirm this. – 2017-02-20
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    Ok. I am using the definition from Chapter 3. I don't quite see how you gathered the correct definition from the preceding theorem but I'll take it. Thanks! – 2017-02-20

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