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I am reading a proof of Serre's criterion for normality that takes for granted the following fact:

Let $I \leq J$ two ideals of a Noetherian ring $R$. If $I_p = J_p$ for every associated prime $p$ of $I$, then $I=J$.

Can someone help me out on how to argue this?

Hammerhead
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  • By chance is it the case that the inclusion $I\subseteq J$ should be reversed, or that we should be considering the associated primes of $J$ instead? – Alex Mathers Nov 29 '19 at 07:24
  • it is stated this way. – Hammerhead Nov 29 '19 at 07:55
  • Can you please provide a reference to the proof? – Zeek Nov 29 '19 at 15:18
  • Agree with Isaac, I'd be interested in reading the proof you're looking at; I'm pretty sure this isn't even true as stated because you could take $I$ to be the zero ideal and $J$ nonzero, but then $I$ has no associated primes so the statement is vacuously true – Alex Mathers Nov 29 '19 at 19:18
  • Unfortunately, these are personal notes from a course I took many years ago. Alex, I don't understand your comment: in a Noetherian ring, even the zero ideal has primary decomposition, right? – Hammerhead Nov 30 '19 at 04:03

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