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The krull principal ideal theorem I am interested in is the one involving that over Noetherian ring, minimal primes over the principal ideal must be height at most 1.

In the proof, it involves

  1. symbolic power of prime ideals

  2. localization at the same prime ideal such that symbolic power of prime ideals is identified as extension of powers of the prime ideal

  3. Quotienting out the principal ideal results in artinian ring

  4. Nakayama Lemma shows any $Q'$ such that $Q'\subset Q\subset P$ where $P$ minimal over $(x)$, $Q$ and $Q'$ are primes, $Q'=0$.

I could reproduce the proof of PIT. I think 4 is necessary for noetherian ring. $2$ is consequence of $1$. $2$ combining $3$ deduces $4$.

Q1. Was there alternative versions of proof which does not invoke symbolic power of prime ideals? I have seen Eisenbud, Atiyah's proof on PIT and they all use symbolic power of prime ideals. The usage of symbolic power of prime ideals is too much ad hoc.

Q2. Symbolic power of prime ideal is more or less generalization of primary ideal. What is the motivation for adopting it and adapting it to the proof?

Q3. What are usage of symbolic power of prime ideals? I have not encountered a problem that needs to use it unless the problem requires PIT application which will quote symbolic power.

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    $\begingroup$ One can prove Krull by using Poincare series; see the last chapter of Atiyah & Macdonald. $\endgroup$ – Lord Shark the Unknown Aug 29 '17 at 17:45
  • $\begingroup$ Late to the party: In Kemper‘s book, it is proven using properties of Artinianity. $\endgroup$ – Kezer Jul 16 at 5:10

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