7

In Atiyah-MacDonald, we have the following theorem (p. 41):

Proposition 3.11.

i) Every ideal in $S^{-1}R$ is an extended ideal.

ii) If $I$ is an ideal in $R$ then $I^{ec} = \bigcup_{s \in S} (I : \langle s \rangle )$. Hence $I^e = (1) = S^{-1}R$ if and only if $I$ meets $S$.

iii) $I = I^{ec}$ if and only if no element of $S$ is a zero-divisor in $R/I$.

iv) The prime ideals of $S^{-1}R$ are in one-to-one correspondence with the prime ideals of $R$ which don't meet $S$.

(I'm omitting point v) of the theorem since my question is about ii),iii) and iv). )

While I am able prove this theorem I'm wondering about how I'll be able to remember it, in particular, statements ii)-iv). Can anyone give me an example of where I'll be using either one of these three statements or all of them?

Thanks.

Rudy the Reindeer
  • 46,359

2 Answers2

5

(too long to be a comment) I don't have a copy of Atiyah-MacDonald in front of me, so I can't currently give exact references. And I wouldn't be surprised if (iii) was mentioned just to give a nice result of (ii) and to facilitate the proof of the (very useful) part (iv).

But when I read (ii)-(iv), the one that jumps out to me in utility is (iv). For example, in proving things about integral extensions, the going-up and going-down theorems, and some forms of Hilbert's Nullstellensatz (all of which are in AM), one uses (iv).

A sort of general pattern will appear. Some rings might be very complicated, so to discern something about their structure we might localize at some prime ideal or another. This is well-behaved, and localizations are often nicer than the original ring itself. This is the pattern used, if I recall correctly, in the proofs of the going-up and going-down theorems in AM.

It might be even better advice to sit down and go through the exercises. AM is known for having many nontrivial results in the exercises, and I'm confident that this proposition will be useful there.

davidlowryduda
  • 91,687
  • Thank you. Going-up/down theorems was exactly the sort of thing that I was looking for. I think these appear 20 pages later in the book. I'm very happy to know that I'll be using this seemingly not so interesting proposition very soon. – Rudy the Reindeer Jun 15 '12 at 13:04
  • I'll wait for a bit with accepting. Perhaps I get some more examples. – Rudy the Reindeer Jun 15 '12 at 13:05
5

Of all of the statements above, number 4 is the one that I've used the most. Let me tell you about what number 4 can be used for:

  1. Proving the lying over theorem: Given a finite extension $A \subset B$, for every prime ideal $P \subset A$ there exists a prime ideal $Q$ of $B$ lying over $P$, i.e. $Q^{c} = P$. You can prove this by noticing that $S^{-1}B$ is a finitely generated $S^{-1}A$ - module (To prove this bit, I remember using some tensor products iirc) and then if you draw an appropriate diagram you can apply (iv). I believe you have Nakayama's Lemma available to you as $S^{-1}B$ is a finitely generated $S^{-1}A$ - module.

  2. Proving that a ring $A$ is absolutely flat iff $A_\mathfrak{m}$ for each maximal ideal $\mathfrak{m}$.

  3. Proving that every prime ideal of $A$ is maximal iff $A/\mathfrak{R}$ is absolutely flat. $\mathfrak{R}$ is the nilradical of $A$.

  4. Atiyah Macdonald problem 3.6 - For this problem one of the localisations iirc reduces to the case of a local ring, and that is very powerful!

  5. Proving that if ring $A$ has no nilpotent elements and $\mathfrak{p}$ a minimal prime ideal of $A$, then $A_\mathfrak{p}$ is a field. You can see a proof here.

  6. This isn't really about the ideal correspondence in (iv) but about the following result: If you have an ideal $I$ disjoint from $S$, then you can always find a prime ideal $P$ such that $P \supset I$ and is maximal with respect to the property that $P \cap S = \emptyset$. This is Krull's lemma, and you can use it to prove problems 1.14 and 3.7(i) of Atiyah - Macdonald.

  7. Proving that the nilradical of $S^{-1}A$ is $S^{-1}\mathfrak{R}$. This is used in the first step in the "if" direction of #3 above.

Community
  • 1