# Atiyah-Macdonald Ex8.6

Is there anybody can give a proof? I can prove "finite" only, but I cannot prove "bounded".

Here is the exercise: Let A be a Noetherian ring and Q a P-primary ideal in A. Consider chains of primary ideals from Q to P. Show that all such chains are of finite bounded length, and that all maximal chains have the same length.

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Hint: consider the (primary) ideals of the ring $A_P / Q A_P$. – Andrea Aug 26 '11 at 7:52
Dear Andrea, I've already considered this. In fact, I proved "finite" by this method. What I can't prove is "bounded", which means the bound is independt of the prime ideal P. – Chan Yat-Fay Aug 26 '11 at 9:19
See my answer below. By the way, the lenght of maximal chains depends on the prime ideal $P$! – Andrea Aug 26 '11 at 10:18

## 2 Answers

Let $A$ a noetherian ring, let $P$ be a prime ideal of $A$ and $Q$ a $P$-primary ideal. Consider the ring $B = A_P / Q A_P$. Prove:

1) Every ideal of $B$ is $\bar{P}$-primary.

2) There is a 1:1 correspondence between the set of $P$-primary ideals of $A$ that contain $Q$ and the set of ideals of $B$. This correspondence preserves containments.

3) $B$ is an artinian ring, so $B$ is a $B$-module of finite length.

4) Every chain of ideals of $B$ has lenght $\leq l_B(B)$. Every maximal chain of ideals of $B$ has lenght $= l_B(B)$. ($l_B(B)$ is the length of $B$ as a $B$-module.)

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Thank you. Your proof just show the finiteness. I also did this. But the exercise asks us to prove not only the finiteness, but also the boundedness. It means that there exists an integer M such that l(B)<M for any prime ideal of A. Do you agree with me? – Chan Yat-Fay Aug 26 '11 at 10:28
You are misinterpreting the text of the exercise. It is false what you are saying: consider in $k[x]$ the chain $(x) \supsetneqq (x^2) \supsetneqq (x^3) \supsetneqq \cdots$. – Andrea Aug 26 '11 at 11:43
Very elegant, Andrea. Especially 1), which eliminates any further consideration of "primary"! A proposito, it might be reminded that an ideal with maximal nilradical is primary. – Georges Elencwajg Aug 26 '11 at 13:27
Thank you for your example, dear @Andrea. It make me realized that I was misinterpreting. – Chan Yat-Fay Aug 26 '11 at 13:46

EDIT Dear User, this is just to tell you that I have trimmed a bit the Google list. It contains all the solutions to the AM exercises I have been able to find on the web, but I'm sure there are many others. If you know some of these others, please add them to this community wiki answer by editing it. TIDE

This is not an answer, and this is a community wiki.

I don't know if it's a good idea, but if you're interested you can help me collect here links to online solutions to Atiyah-MacDonald exercises.

Here is my Google list of such links.

It's very messy, and some links are broken.

If it turn out to be a bad idea, I can delete this "answer", and give the link in a comment.

Here is a MathOveflow Errata for Atiyah-MacDonald.

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Do you mean this result of this exercise is false? Is there any counter examples? – Chan Yat-Fay Aug 26 '11 at 9:23
Dear @Chan: No, I don't mean that. My "answer" is not really an answer. It has nothing to do with Exercise 8.6. It's only related to the exercises of this book in general. Sorry if I wasn't clear. – Pierre-Yves Gaillard Aug 26 '11 at 9:31
OK, I see. And your idea is good, I think. Thank you! – Chan Yat-Fay Aug 26 '11 at 10:07
Hi Pierre-Yves, do you have any other google lists for other books? – yunone Aug 27 '11 at 6:42
Hi @yunone! No, sorry... (BTW I've just organized the google list. It's slightly less messy now. Thanks for your interest.) - To the other: from yunone's MSE page: "A hobby of mine is attempting to 'completely solve' books as I go through them. I'm currently working on the following: Herbert Enderton's Elements of Set Theory; Kenneth Ireland and Michael Rosen's Modern Number Theory..." – Pierre-Yves Gaillard Aug 27 '11 at 7:00