Rings, Modules, Fields

8 Valuation rings

Definition 8.1. An ordered abelian group is an abelian group \(\Gamma \) with a total ordering \(\leq \) that is translation invariant: \(a\leq b \iff a+x \leq b+x\).

Lemma 8.2. An ordered abelian group is determined by a collection of positive elements closed under addition that have trichotomy: every element is either positive, negative, or \(0\).

Proof. Given an ordered abelian group, the positive elements certainly are closed under addition, and satisfy trichotomy. Conversely, we can define \(a \geq b \iff a-b \geq 0\). Transitivity is equivalent to being closed under addition, totality is equivalent to trichotomy. □

Lemma 8.3. An abelian group can be ordered iff it is nontorsion.

Proof. Any ordering must restrict to subgroups, and it is easy to see that \(\ZZ /n\ZZ \) has no orderings for \(n>1\), giving one direction. In the other direction, if \(M\) is nontorsion, \(M\otimes \QQ \) is a \(\QQ \)-vector space so has an ordering, and this restricts to one on \(M\). □

Definition 8.4. A complete ordered abelian group is one where any set of positive elements has an infimum.

The forgetful functor from complete ordered abelian groups to ordered abelian groups has a left adjoint, the Dedekind completion. To construct it for a given \(G\), call a subset \(I \subset G\) a Dedekind cut if it is bounded below, and \(a \in I, a\leq b \implies b \in I\). We can turn the collection of Dedekind cuts into an ordered abelian group having \(I+J\) be the set of elements that are sums of elements in \(I\) and \(J\). The inverse of \(I\) is \(\{x|x+I \geq 0\}\). The inclusion \(g \mapsto \{x|x\geq g\}\) is the universal map, and the infimum is given by the union.

Lemma 8.5. A well ordered abelian group is either trivial or \(\ZZ \).

Proof. If it is nontrivial, let \(a\) be a positive element. WLOG, we can assume that for \(n \in \NN \), there is no \(x\) such that \(nx = a\), as otherwise choose \(a = x\). If we could do this process infinitely many times, than there would be an infinite subset \(X\) of \(\NN \) such that \(\frac a n, n \in X\) is in the abelian group, but then this set would have no minimum. Now If \(\langle a \rangle \) is not everything, then suppose \(b\) is another positive element. \(\langle a,b\rangle = \ZZ ^2\). WLOG we can assume \((1,-1)\) is positive. But then the positive elements of the form \((a,x), a>0\) and \((0,x),x>0\), which is not a well ordering. □

Definition 8.6. A valuation ring is an integral domain \(R\) with a valuation: a surjective homomorphism \(\nu :\Frac (R)^\times \to \Gamma \), where \(\Gamma \) is an ordered abelian group such that \(R\) is preimage of the non-negative elements.

Note that in a valuation ring, \(a|b\) iff \(\nu (a) \leq \nu (b)\). In particular, two nonzero elements are associates iff they have the same valuation, and so units of \(R\) are those with valuation \(0\).

Proposition 8.7. TFAE:
- 1. \(R\) is a valuation ring.
- 2. Either \(x\) or \(x^{-1}\) is in \(R\) for each \(0 \neq x \in \Frac (R)\).
- 3. \(R\) is a domain and the (principle) ideals are totally ordered by inclusion.

Proof. \(1 \implies 3\) since ideals correspond to Dedekind cuts of \(\Gamma \) via the valuation. Note that principle ideals correspond to elements of \(\Gamma \) in the completion. \(3 \implies 2\) as if \(a,b \in R\), then \((a)\subset (b)\) or \((b) \subset (a)\) so that \(\frac a b\) or \(\frac b a \in R\). \(2 \implies 1\), as the projection to \(\Frac (R)^\times /R^\times \) is a valuation, where the non-negative elements are the image of \(R_z\). □

Note as a consequence that \(\nu (a+b) \geq \min (\nu (a),\nu (b))\) if \(a,b,a+b \neq 0\), and that the valuation is essentially unique.

Corollary 8.8. A valuation ring is an integrally closed local Bezout domain of dimension at most \(1\).

Proof. There can only be one maximal ideal as the ideals are totally ordered, and since finite sets have minima it is Bezout by the remark in the previous proof. If \(0 \neq x \in \Frac (R)\) is integral over \(R\), WLOG the polynomial it is a root of has a nonzero constant term. Then that polynomial times □

valuation rings are “maximal".

If the valuation ring \(R\) is Noetherian, then the set of positive elements must be well ordered, as given a set of positive elements, the generator of the ideal they generate is principle, so any set of positive elements has a lower bound contained in the set. Thus either the valuation is trivial, in which case the ring is a field, or its image is \(\ZZ \), in which case we say \(R\) is a discrete valuation ring or DVR.