Proving Artinian of Commutative Noetherian Rings with Maximal Primes

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Discussion Overview

The discussion revolves around proving that a commutative Noetherian ring in which all prime ideals are maximal is Artinian. The scope includes theoretical reasoning and mathematical proof techniques.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested, Mathematical reasoning

Main Points Raised

  • One participant proposes that a descending chain of prime ideals leads to the conclusion that the ring is Artinian, stating that since all primes are maximal, the chain must stabilize.
  • Another participant questions the validity of focusing solely on descending chains of prime ideals, pointing out that this does not necessarily imply the ring is Artinian, referencing the example of the integers.
  • The same participant challenges the conclusion that I_n = I_{n+1} follows from maximality, suggesting that this reasoning is flawed.
  • A side note mentions that while every commutative Noetherian ring satisfies the descending chain condition on prime ideals, this does not imply that all such rings are Artinian.

Areas of Agreement / Disagreement

Participants express disagreement regarding the sufficiency of the proposed proof and the implications of maximality on the behavior of prime ideals. The discussion remains unresolved as no consensus is reached on the correctness of the initial claim.

Contextual Notes

There are limitations in the reasoning presented, particularly regarding the assumptions about maximal ideals and the implications for Artinian properties. The discussion highlights the need for further clarification on the relationship between prime ideals and Artinian conditions.

peteryellow
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prove that a commutative noetherian ring in which all primes are maximal is artinian.
 
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So, what have you tried?
 
well I don't have an idea how to start...
 
Let there be given a decending chain of prime ideals I_1 \supset I_2 \supset I_3 \supset I_4 \supset...Since all primes are maximal therefore for a natural number n we have I_n =I_{n+1}. Hence the ring is artinian.

Is it correct? Please help thanks.
 
Why is it sufficient to look at descending chains of prime ideals? Is it true that if a ring R satisfies the descending chain condition on its prime ideals then R is Artinian? (No: take R=[itex]\mathbb{Z}[/itex].) Also, how did you conclude that I_n = I_{n+1}? This doesn't follow from maximality.

Try again!

[Side note: Incidentally, one can prove that every commutative Noetherian ring satisfies the descending chain condition on prime ideals. So if this were sufficient to determine if a ring is Artinian, then we would be able to conclude that every commutative Noetherian ring is Artinian, which is definitely not the case.]
 
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