Posted in math.AC, math.AG, math.GN, tagged MaBloWriMo on November 28, 2009|
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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.

Akhil’s posts at Delta Epsilons here and here describe the basic properties of Noetherian rings well, including the proof of the following.

**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.

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