Today’s (quick) topic focuses on Dedekind domains. These come up when you take the ring of integers in any finite extension of (i.e. number fields). In these, you don’t necessarily have unique factorization. But you do have something close, which makes these crucial.
Definition 1 A Dedekind domain is a Noetherian integral domain that is integrally closed, and of Krull dimension one—that is, each nonzero prime ideal is maximal.
A DVR is a Dedekind domain, and the localization of a Dedekind domain at a nonzero prime is a DVR by this. Another example (Serre) is to take a nonsingular affine variety of dimension 1 and consider the ring of globally regular functions ; the localizations at closed points are DVRs, so the ring is a Dedekind domain.
Now assume is Dedekind.
A f.g. -submodule of the quotient field is called a fractional ideal; by multiplying by some element of , we can always pull a fractional ideal into , when it becomes an ordinary ideal. The sum and product of two fractional ideals is a fractional ideal.
Theorem 2 (Invertibility) If is a nonzero fractional ideal and , then is a fractional ideal and .
Thus, the nonzero fractional ideals are an abelian group under multiplication.
To see this, note that invertibility is preserved under localization: for a multiplicative set , we have , where the second ideal inverse is with respect to ; this follows from the fact that is finitely generated. Note also that invertibility is true for discrete valuation rings.
So for all primes , we have , which means the inclusion of -modules is an isomorphism at each localization. Therefore it is an isomorphism, by general algebra.
The next result says we have unique factorization of ideals:
Theorem 3 (Factorization) Each ideal can be written uniquely as a product of powers of prime ideals.
Let’s use the pseudo-inductive argument. Let be the maximal ideal which can’t be written in such a manner, since is Noetherian. Then isn’t prime, so it’s contained in some prime . But , and can be written as a product of primes, contradiction.
Uniqueness follows by localizing at each prime.