Commutative Algebra/Valuation rings

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Augmented ordered Abelian groups[edit]

In this section, for reasons that will become apparent soon, we write Abelian groups multiplicatively.

Definition 18.1:

An ordered Abelian group is a group together with a subset such that:

  1. is closed under multiplication (that is, ).
  2. If , then . (This implies in particular that .)
  3. .

We write ordered Abelian groups as pair .

The last two conditions may be summarized as: is the disjoint union of , and .

Theorem 18.2:

Let an ordered group be given. Define an order on by

, .

Then has the following properties:

  1. is a total order of .
  2. is compatible with multiplication of (that is, and implies ).

Proof:

We first prove the first assertion.

is reflexive by definition. It is also transitive: Let and . When or , the claim follows trivially by replacing in either of the given equations. Thus assume and . Then and hence (even ).

Let and . Assume for a contradiction. Then and , and since is closed under multiplication, , contradiction. Hence .

Let such that . Since , (which is not equal ) is either in or in (but not in both, since otherwise and since , , contradiction). Thus either or .

Then we proceed to the second assertion.

Let . If , the claim is trivial. If , then , but . Hence .

Definition 18.3:

Let be an ordered Abelian group. An augmented ordered Abelian group is together with an element (zero) such that the following rules hold:

, .

We write an augmented ordered Abelian group as triple .

Valuations and valuation rings[edit]

Definition 18.4:

Let be a field, and let be an augmented ordered Abelian group. A valuation of the field is a mapping such that:

  1. .
  2. .
  3. .

Definition 18.5:

A valuation ring is an integral domain , such that there exists an augmented ordered Abelian group and a valuation with .

Theorem 18.6:

Let be a valuation ring, and let be its field of fractions. Then the following are equivalent:

  1. is a valuation ring.
  2. is an integral domain and the ideals of are linearly ordered with respect to set inclusion.
  3. is an integral domain and for each , either or .

Proof:

We begin with 3. 1.; assume that

1. 2.: Let any two ideals. Assume there exists . Let any element be given.

Properties of valuation rings[edit]

Theorem 18.8:

A valuation ring is a local ring.

Proof:

The ideals of a valuation ring are ordered by inclusion. Set . We claim that is a proper ideal of . Certainly for otherwise for some proper ideal of . Furthermore, .

Theorem 18.9:

Let be a Noetherian ring and a valuation ring. Then is a principal ideal domain.

Proof:

For, let be an ideal; in any Noetherian ring, the ideals are finitely generated. Hence let . Consider the ideals of . In a valuation rings, the ideals are totally ordered, so we may renumber the such that . Then .