Finite Prime Ideals in Noetherian Ring - Atiyah-McDonald

[disregardthat]

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?)

 

[micromass]

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...

 

[disregardthat]

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.