Editing Valuation ring (section)

== Definitions ==
There are several equivalent definitions of valuation ring (see below for the characterization in terms of dominance). For an [[integral domain]] ''D'' and its [[field of fractions]] ''K'', the following are equivalent:
# For every non-zero ''x'' in ''K'', at least one of ''x'' or ''x''<sup>−1</sup> is in ''D''.
# The ideals of ''D'' are [[totally ordered]] by inclusion.
# The principal ideals of ''D'' are [[totally ordered]] by inclusion (i.e. the elements in ''D'' are, [[up to]] [[unit (ring theory)|units]], totally ordered by [[divisibility (ring theory)|divisibility]].)
# There is a [[totally ordered group|totally ordered]] [[abelian group]] Γ (called the '''value group''') and a '''[[valuation (algebra)|valuation]]''' ν: ''K'' → Γ ∪ {∞} with ''D'' = { ''x'' &isin; ''K'' | ν(''x'') ≥ 0 }.

The equivalence of the first three definitions follows easily. A theorem of {{harv|Krull|1939}} states that any [[ring (mathematics)|ring]] satisfying the first three conditions satisfies the fourth: take Γ to be the [[quotient group|quotient]] ''K''<sup>×</sup>/''D''<sup>×</sup> of the [[unit group]] of ''K'' by the unit group of ''D'', and take ν to be the natural projection. We can turn Γ into a [[totally ordered group]] by declaring the residue classes of elements of ''D'' as "positive".{{efn|More precisely, Γ is totally ordered by defining <math>[x] \geq [y]</math> [[if and only if]] <math>x y^{-1} \in D</math> where [''x''] and [''y''] are equivalence classes in Γ. cf. {{harvp|Efrat|2006|p=39}}}}

Even further, given any totally ordered abelian group Γ, there is a valuation ring ''D'' with value group Γ (see [[Hahn series]]).

From the fact that the ideals of a valuation ring are totally ordered, one can conclude that a valuation ring is a local domain, and that every finitely generated ideal of a valuation ring is principal (i.e., a valuation ring is a [[Bézout domain]]). In fact, it is a theorem of Krull that an integral domain is a valuation ring if and only if it is a local Bézout domain.{{sfn|Cohn|1968|loc=Proposition 1.5}} It also follows from this that a valuation ring is [[Noetherian ring|Noetherian]] if and only if it is a [[principal ideal domain]]. In this case, it is either a field or it has exactly one non-zero prime ideal; in the latter case it is called a [[discrete valuation ring]]. (By convention, a field is not a discrete valuation ring.)

A value group is called ''discrete'' if it is [[group isomorphism|isomorphic]] to the additive group of the [[Integer#Algebraic_properties|integers]], and a valuation ring has a discrete valuation group if and only if it is a [[discrete valuation ring]].{{sfn|Efrat|2006|p=43}}

Very rarely, ''valuation ring'' may refer to a ring that satisfies the second or third condition but is not necessarily a domain.  A more common term for this type of ring is ''[[serial module|uniserial ring]]''.