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Given a Noetherian ring $R$ and a proper ideal $I$ of it, is it true that $\cap_n I^n=0$ as $n$ varies over all natural numbers. If not, is it true if $I$ is a maximal ideal? If not, is it true if $I$ is the maximal ideal of a local ring $R$. If not, is it true under additional assumptions on $R$ (like $R$ is regular)

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Dear Dev, it is true if $I$ is the maximal ideal of a noetherian local ring $R$. This is the Krull intersection theorem, and follows quickly from a combination of the Artin-Rees lemma and Nakayama's lemma. –  Akhil Mathew Jan 25 '11 at 2:48
    
@Akhil: Thanks a lot. That helps. Do you know if it's true in the regular case if $R$ is not local. –  Dev Bappa Jan 25 '11 at 3:11
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A ring of the form $k \times k$ is certainly regular. –  Pete L. Clark Jan 25 '11 at 3:46
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2 Answers

up vote 6 down vote accepted

It is not true in general: the ideal may well be idempotent!

For an example, consider a direct product of two fields: there are two ideals, both maximal and both idempotent.

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Thanks for that example. –  Dev Bappa Jan 25 '11 at 3:12
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See the Section on the Krull Intersection Theorem (currently Section 8.7) in these notes.

A version of the theorem valid for any ideal $I$ in a Noetherian ring $R$ is as follows: if there exists $x \in \bigcap_{n=1}^{\infty} I^n$, then $x \in xI$. From this one easily deduces that $\bigcap_{n=1}^{\infty} I^n = \{0\}$ under either of the following additional hypotheses:

$\bullet$ $R$ is a domain and $I$ is a proper ideal, or
$\bullet$ $I$ is contained in the Jacobson radical $J(R)$ of $R$ (i.e., the intersection of all maximal ideals).

In particular the second condition holds for any proper ideal in a Noetherian local ring.

As Mariano remarks, some hypothesis beyond Noetherianity is needed in order to guarantee $\bigcap_n I^n = \{0\}$. I should probably add his counterexample to my notes!

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you should hyperref your notes! –  Mariano Suárez-Alvarez Jan 25 '11 at 4:01
    
@Mariano: I am too embarrassed to admit that I don't even know what that means, but perhaps someone will explain anyway. –  Pete L. Clark Jan 25 '11 at 4:12
    
it is impossible to be sure without seeing your latex file, but if you say \usepackage{hyperref} late in the prologue of your file (say, after all other packages have been loaded), there are great chances that the resulting PDF file will have all its internal cross references turned into working hyperlinks. –  Mariano Suárez-Alvarez Jan 25 '11 at 4:19
    
@Mariano: I tried what you said, and I did not get any hyperlinks in my pdf file. But I appreciate the suggestion, and when I get the chance I'll ask around and find out how to make it work. –  Pete L. Clark Jan 25 '11 at 8:02
    
if you point me to the source file, I can take a look; email, e.g. –  Mariano Suárez-Alvarez Jan 25 '11 at 8:04
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