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I'm trying to solve the exercise 6.7 of Miles Reid's Undergraduate Commutative Algebra (pag 93).

How can I prove that if $B$ is a finite ring extension of $A$, there are only finitely many prime ideals of $B$ whose intersection with $A$ is a given prime ideal of $A$?

Thank you!

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marked as duplicate by user26857, Claude Leibovici, Dario, SHOBHIT GAUTAM, Surb Feb 28 '15 at 16:20

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

    
This is also the exercise 9.3 in Matsumura, Commutative Ring Theory. – user26857 Mar 17 '13 at 20:53
up vote 3 down vote accepted

Hint: Let $\mathfrak{p} \subset A$ be a prime ideal. Localize at $\mathfrak{p}$, so that $A_{\mathfrak{p}}$ is a local ring. The primes $\mathfrak{q}$ in $B$ for which $\mathfrak{q} \cap A = \mathfrak{p}$ are called primes lying over $\mathfrak{p}$. The primes lying over $\mathfrak{p}$ remain in the ring $B_{\mathfrak{p}},$ and are distinct. All these primes lie over the maximal ideal $\mathfrak{p}A_{\mathfrak{p}}$ in $A_{\mathfrak{p}}$, so they are all maximal. Now, how many maximal ideals containing $\mathfrak{p}B_{\mathfrak{p}}$ can $B_{\mathfrak{p}}$ have?

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I was trying to prove that $B_p/pB_p$ is an Artinian Ring, so it has a finite number of maximal ideals. Hence $B_p$ has only finitely many maximal ideals containing $pB_p$. Is this the right way? – Corra Jan 4 '13 at 19:06
    
Could you suggest me a way to prove that $B_p/pB_p$ is Artinian? – Corra Jan 4 '13 at 19:08
    
Thank you very much, I should have seen it by myself but I'm not very familiar with Artinian rings. Very useful! – Corra Jan 4 '13 at 20:23

Hint: First show that it's enough to prove this for maximal $\mathfrak p$ (localize). Then show that it's enough to prove this when the maximal ideal is $(0)$ (factor). Then you have a finite $k$-algebra and must show that it has finitely many prime ideals.

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