Valuation Ring - Definitions

Definitions

There are several equivalent definitions of valuation ring. For a subring D of its field of fractions K the following are equivalent:

• For every nonzero x in K, either x in D or x−1 in D
• The ideals of D are totally ordered by inclusion
• The principal ideals of D are totally ordered by inclusion
• There is a totally ordered abelian group G (called the value group) and a surjective group homomorphism (called the valuation) ν:K×→G with D = { x in K× : ν(x) ≥ 0 } ∪ {0}

The equivalence of the first three definitions follows easily. A theorem of (Krull 1939) states that any ring satisfying the first three conditions satisfies the fourth: take G to be the quotient K×/D× of the unit group of K by the unit group of D, and take ν to be the natural projection. Even further, given any totally ordered abelian group G, there is a valuation ring D with value group G.

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 "uniserial ring".