Let’s think more about what an abstract theory of unique factorization of primes has to look like. One fundamental property it has to satisfy is that factorizations should be finite. Another way of saying this is that the process of writing elements as products of other elements (up to units) should end in a finite set of irreducible elements at some point. This condition is clearly not satisfied by sufficiently “large” commutative rings such as , the ring of fractional polynomials.
Since we know we should think about ideals instead of numbers, let’s recast the problem in a different way: because we can write for any , the ascending chain of ideals never terminates. In any reasonable theory of factorization writing and then comparing the ideals , then repeating this process to obtain a chain of ideals , should eventually stabilize at a prime. This leads to the following definition.
Definition: A commutative ring is Noetherian if every ascending chain of ideals stabilizes.
Hilbert’s Basis Theorem: If is a Noetherian ring, so is .
Today we’ll discuss what the Noetherian condition means in terms of the topology of . The answer turns out to be quite nice.
The Noetherian condition is actually so strong that not every UFD satisfies it! For example, the ring of polynomials in countably many variables doesn’t satisfy the Noetherian condition because doesn’t stabilize, but this ring is clearly a UFD because the factors of any given polynomial lie in a finitely-generated subring. But from a geometric perspective we don’t really care about such rings, since contains . From a practical perspective any question concerning finitely many elements of this ring actually happens in a Noetherian subring.
As we saw above, the Noetherian condition implies that every non-zero non-unit is a product of irreducibles, although this representation is not necessarily unique. It follows that a Noetherian integral domain is a UFD if and only if every irreducible element is prime.
The Noetherian condition is equivalent to the condition that every ideal of is finitely generated, so it follows that PIDs are Noetherian, hence are Noetherian. Hilbert’s basis theorem together with a few other basic facts then tell us that any finitely-generated algebra over a Noetherian ring is Noetherian, so anything we want to do with polynomial rings or number rings will stay in the Noetherian regime.
Remember that when we wanted to characterize the maximal ideals of the ring of continuous functions on a compact Hausdorff space, we needed the compactness condition to take an ideal not contained in the ideals and extract finitely many elements of it. The Noetherian condition automatically lets us do this, so it’s reasonable to suppose that the Noetherian condition is equivalent to a kind of compactness. In fact, more is true.
Proposition: If a commutative ring is Noetherian, then every subset of is compact in the Zariski topology, i.e. is hereditarily compact, also known as Noetherian.
Proof. Let . It is a general topological fact that to show that every open cover of a space has a finite subcover it suffices to do so for open covers by a basis of the topology, which here is given by the sets . So let be such an open covering of . The ideal generated by all the is finitely generated, say by . For any point there is some such that , so the same must be true for the generators , and this is equivalent to being a finite subcover.
Unfortunately, the converse is false. For example, let . The “evaluation” homomorphism sending a power series to its constant term has kernel the maximal ideal , and since any element of is invertible in , it follows that is the unique maximal ideal, so consists of a single point, i.e. is a local ring. But the ascending chain of ideals does not stabilize. (I believe replacing with , the prime spectrum, fixes this equivalence, but we won’t worry too much about the prime spectrum for now.)
Corollary: If is Noetherian and is Hausdorff, then is finite.
I sort of glossed over why we know that is non-empty. To prove this we need to know that every ideal of is contained in a maximal ideal. For general rings with unity this is a consequence of Zorn’s lemma, which I am reticent to apply since it is equivalent to the axiom of choice. However, for Noetherian rings one can get by with the axiom of countable choice, which is yet another reason to like Noetherian rings.
Next time we’ll see how the Noetherian condition allows us to decompose a variety into components.