Commutative Algebra/Valuation rings

Augmented ordered Abelian groups In this section, for reasons that will become apparent soon, we write Abelian groups multiplicatively. 0 100% #FFFFFF Definition 18.1: 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 . 0 100% #FFFFFF Theorem 18.2: 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 ). 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 . 0 100% #FFFFFF 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 0 100% #FFFFFF 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: . . . 0 100% #FFFFFF Definition 18.5: A valuation ring is an integral domain , such that there exists an augmented ordered Abelian group and a valuation with . 0 100% #FFFFFF Theorem 18.6: 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 . 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 0 100% #FFFFFF 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, . 0 100% #FFFFFF 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 .