Given a field F, if D is a subring of F such that either x or x −1 belongs to D for every x in F, then D is said to be a valuation ring for the field F or a place of F. Since F is in this case indeed the field of fractions of D, a valuation ring for a field is a valuation ring. Another way to characterize the valuation rings of a field F is that valuation rings D of F have F as their field of fractions, and their ideals are totally ordered by inclusion; or equivalently their principal ideals are totally ordered by inclusion. The valuation rings of a field are the maximal elements of the local subrings partially ordered by dominance, where
- dominates if and .
In particular, every valuation ring is a local ring. An integral domain whose localization at any prime ideal is a valuation ring is called a Prüfer domain.
Other articles related to "valuation ring, ring, valuation":
... In abstract algebra, a discrete valuation ring (DVR) is a principal ideal domain (PID) with exactly one non-zero maximal ideal ... R is a valuation ring with a value group isomorphic to the integers under addition ... R is a noetherian local ring with Krull dimension one, and the maximal ideal of R is principal ...
... numbers and v be its usual p-adic valuation (with ) ... is a (not necessarily algebraic) extension field of K, itself equipped with a valuation w, we say that is formally p-adic when the following conditions are satisfied w extends v (that is, for all x in K), the ... the field ℚ(i) of Gaussian rationals, if equipped with the valuation w given by (and ) is formally 5-adic (the place v=5 of the rationals splits in two places of the Gaussian rationals since ...
... A PID with only one non-zero maximal ideal is called a discrete valuation ring, or DVR, and every discrete valuation ring is a valuation ring ... A valuation ring is a PID if and only if it is a DVR or a field ... if and only if it is isomorphic to the additive group of the integers, and a valuation ring has a discrete valuation group if and only if it is a discrete valuation ring ...
Famous quotes containing the word ring:
“Roll unmanly over this turning tuft,
O ring of seas, nor sorrow as I shift
From all my mortal lovers with a starboard smile....”
—Dylan Thomas (19141953)