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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??

user26857
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user7090
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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. – tracing Feb 20 '17 at 02:02
  • 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! – user7090 Feb 20 '17 at 02:10
  • @user7090 In the paragraph right after the definition of associated primes, Eisenbud writes that the associated primes of an ideal $I$ will be understood to mean the associated primes of $R/I$. – Juan L. Dec 09 '20 at 21:47

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In the paragraph right after the definition of associated primes, Eisenbud writes that the associated primes of an ideal $I$ will be understood to mean the associated primes of $R/I$.

Juan L.
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