Augmented ordered Abelian groups
In this section, for reasons that will become apparent soon, we write Abelian groups multiplicatively.
An ordered Abelian group is a group together with a subset such that:
- is closed under multiplication (that is, ).
- If , then . (This implies in particular that .)
We write ordered Abelian groups as pair .
The last two conditions may be summarized as: is the disjoint union of , and .
Let an ordered group be given. Define an order on by
- , .
Then has the following properties:
- is a total order of .
- is compatible with multiplication of (that is, and implies ).
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 .
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
Let be a field, and let be an augmented ordered Abelian group. A valuation of the field is a mapping such that:
A valuation ring is an integral domain , such that there exists an augmented ordered Abelian group and a valuation with .
Let be a valuation ring, and let be its field of fractions. Then the following are equivalent:
- is a valuation ring.
- is an integral domain and the ideals of are linearly ordered with respect to set inclusion.
- is an integral domain and for each , either or .
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
A valuation ring is a local ring.
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, .
Let be a Noetherian ring and a valuation ring. Then is a principal ideal domain.
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 .