11

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!

Corra
  • 225

2 Answers2

8

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?

Elchanan Solomon
  • 30,807
  • 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 04 '13 at 19:06
  • Could you suggest me a way to prove that $B_p/pB_p$ is Artinian? – Corra Jan 04 '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 04 '13 at 20:23
  • Could you explain how "all these primes lie over $\mathfrak{p}A_\mathfrak{p}$" implies that they are all maximal, and how that implies that they contain $\mathfrak{p}B_\mathfrak{p}$? – Anakhand Oct 13 '23 at 10:07
2

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.

Jim
  • 30,682