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I am trying to prove that the nilradical of a Noetherian ring is the intersection of finitely many prime ideals but can not seem to do it.

I am trying to make a standard argument assuming, for contradiction, that the nilradical is not the intersection of finitely many prime ideals and deriving a contradiction but can't find one, any help is much appreciated.

Learning Representation Theory
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  • which definition are you taking for the nilradical. Do you know it's the intersection of all prime ideals? – N8tron May 30 '18 at 23:21
  • Yes, I was trying to somehow use that. Also, I know how to prove this result using Spec(R) and irreducible components but wanted to prove it using more basic properties of Noetherian rings. – Learning Representation Theory May 30 '18 at 23:33
  • Take the primary decomposition of the nilradical. Show that the primary decomposition of radical ideal consists of primes. The finiteness follows from Noether's theorem. – Youngsu May 31 '18 at 04:58

1 Answers1

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Hint:

The nilradical is the intersection of all minimal prime ideals. So prove a noetherian ring has only a finite number of minimal primes.

Bernard
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