Noetherian Rings


I promised this awhile back. It seems as if the Noetherian condition is really the last major thing I need before being able to move on.

A ring is Noetherian if every ascending chain of ideals stabilizes (or “terminates”). So, this means that given any collection of ideals \{I_n\}\subset R such that I_1\subset I_2\subset I_3 \subset \cdots we have that there exists some N so that I_n=I_{n+1}=\cdots for all n>N. This condition seems very strange at first. It is known as the Ascending Chain Condition, or ACC for short, but it turns out that it is equivalent to some other things and makes sure our rings are somewhat well-behaved.

Since for the purpose of this collection of posts we only care about commutative rings, the ACC is equivalent to the condition that every ideal is finitely generated.

Proof) Suppose every ideal is finitely generated. Then let I_n be an ascending chain of ideals. Since I=\cup I_n is an ideal, it is generated by say m elements: I=<a_1, \ldots, a_m>. But each one of these elements come from some specific ideal, so suppose a_1\in I_{n_1}, \ldots, a_m\in I_{n_m}. Then just take N=\max(n_1, \ldots, n_m) and we have that the chain stabilizes after that.

For the reverse we go by contrapositive. Let I\subset R be some ideal that is not finitely generated. Then we can find a_1\in I such that <a_1>\neq I. We can also find a_2\in I\setminus <a_1> such that <a_1, a_2>\neq I. We can continue this process without termination. If it terminated at some step then, the ideal would be finitely generated. Thus we now just note that we have an ascending chain that doesn’t terminate <a_1>\subset <a_1, a_2>\subset \cdots.

It is easily seen that every PID is Noetherian. Rings tend to stay Noetherian under new constructions. The ring of polynomials (in finitely many indeterminates) and ring of power series where the coefficients come from a Noetherian ring is Noetherian. The former is known as the Hilbert Basis Theorem. Both the quotient R/I and the ring of fractions S^{-1}R are Noetherian if R is Noetherian.

But remember we want to figure out how this works with prime ideals. It turns out that prime isn’t quite what we want to get the best results, but in order to not introduce yet another type of ideal, I’ll leave this out since it won’t appear in anything I do later. So it turns out that if I\subset R an ideal and R Noetherian, every prime ideal P\supset I contains a minimal-over-I prime ideal, say P_0\supset I. This is just a standard one step application of Zorn’s Lemma.

So I think I’ve beat primality to death. Next time I’ll do a sort of “history of math” type post on Hilbert’s Zahlbericht to put into the blog carnival. This will give me some time to think of where to go from here. I’m thinking the algebraic number theory side…I just don’t want to have to build Galois theory before I do it.

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