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I am currently engaged in independent study of algebraic geometry, using Dan Bump's book. One of the exercises in it outlines a proof of the Krull Intersection Theorem, which [here] is the following:

Let $A$ be a Noetherian local ring with maximal ideal $\mathfrak{m}$, and let $M$ be the intersection of all of the $\mathfrak{m}^n$. Then $M = 0$.

The hints direct me to use the Artin-Rees lemma to show that $\mathfrak{m} M = M$, then use Nakayama's lemma to show that $M = 0$ (this second step is easy). I showed this to a professor and he accused the book of using big machinery for no reason, arguing that

$\mathfrak{m} M = \mathfrak{m} \bigcap_{n \ge 0} \mathfrak{m}^n = \bigcap_{n \ge 1} \mathfrak{m}^n = M.$

Does this argument work? Does Bump apply Artin-Rees because that argument works in some broader context where the above argument fails?

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    $(x^2) \cap (x) = (x^2) \neq (x^3)$ which is the product of the ideals inside $\mathbb{Z}[x]$2010-12-05

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There is a proof due to Herstein that is "elementary" in the sense of avoiding the Artin-Rees lemma. Below is Kaplansky's presentation of this proof, from his "Commutative Rings". A proof using primary decomposition (as in Krull's original proof) can be found in Zariski and Samuel. alt text alt text

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See Theorem 3.6 in Chapter 6 of the CRing project for another elementary argument due to Perdry (American Math. Monthly, 2004) using only the Hilbert basis theorem. In fact, it shows more: if $R$ is a noetherian domain, $I \subset R$ a proper ideal, then the intersection of the powers of $I$ is trivial.

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    @Pete: Dear Pete, yes, the project is very unfinished right now and frequently reflects its origins in a set of notes. Thank you for your kind words (and for the link on your website!). We should indeed chat sometime about complementary paths. Right now I (at least) envision the CRing project as taking a heterodox stance for functoriality and universal properties (kind of like the book "Algebraic topology from a homotopical viewpoint"). I also have had great fun surfing old issues of the Monthly to look for material to add.2011-01-30