Finite Prime Ideals in Noetherian Ring - Atiyah-McDonald

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SUMMARY

In a Noetherian ring, every ideal has a finite primary decomposition, which leads to a finite number of minimal prime ideals. This conclusion is supported by Lemmas 7.11 and 7.12 in the Atiyah-Macdonald book. The discussion also clarifies that the minimal prime ideals correspond to those that do not contain any other prime ideal containing the ideal in question. Additionally, it is confirmed that the finite number of prime ideals of height 0 exists for all ideals.

PREREQUISITES
  • Noetherian rings
  • Primary decomposition in ring theory
  • Prime ideals and their properties
  • Atiyah-Macdonald's "Introduction to Commutative Algebra"
NEXT STEPS
  • Study the implications of Lemma 4.6 in Atiyah-Macdonald regarding prime ideals
  • Explore the concept of height of prime ideals in Noetherian rings
  • Investigate the relationship between primary decomposition and minimal prime ideals
  • Review examples of Noetherian rings to solidify understanding of finite prime ideals
USEFUL FOR

Mathematicians, algebraists, and graduate students studying commutative algebra, particularly those focusing on Noetherian rings and prime ideal theory.

disregardthat
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In a noetherian ring, why is it true that there are only a finite number of minimal prime ideals of some ideal? (And is it proven somewhere in the Atiyah-mcdonald book?)
 
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Check out lemma 7.11 and 7.12. It gives you that every ideal has a FINITE primary decomposition. Thus there are also finitely many minimal prime ideals...
 
Yes, I know this, but does these correspond to all prime ideals of a that does not contain any other prime ideal containing a? I meant minimality in this sense.

What I am effectively is asking is that: Does A\a have a finite number of prime ideals of height 0 for all ideals a?

EDIT: Ok, I seem to have missed it, it is proven in 4.6 in the book.
 

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