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Let $R$ be a commutative Noetherian ring with unity. I want to prove the fact that depth $R_{\mathfrak p}=0$ implies $\mathfrak pR_{\mathfrak p}$ is associated to zero.

Since depth $R_{\mathfrak p}=0$ we have that $\mathfrak pR_{\mathfrak p}$ consists of zero divisors. But to prove that it is a prime associated to $0$, I need to show that it is an annihilator ideal of some element in $R_{\mathfrak p}$.

How do I show this?

Thank you

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    $\begingroup$ The set of zero-divisors is the union of associated primes. $\endgroup$ – user26857 May 9 at 15:02
  • $\begingroup$ Yes. You can check Atiyah Prop. 4.7 for a reference $\endgroup$ – pyrogen May 10 at 9:12

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