In abstract algebra, a valuation ring is an integral domain D such that for every element x of its field of fractions F, at least one of x or x −1 belongs to D.
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.
Read more about Valuation Ring: Examples, Definitions, Properties, Units and Maximal Ideals, Value Group, Integral Closure, Principal Ideal Domains
Famous quotes containing the word ring:
“The world,—this shadow of the soul, or other me, lies wide around. Its attractions are the keys which unlock my thoughts and make me acquainted with myself. I run eagerly into this resounding tumult. I grasp the hands of those next to me, and take my place in the ring to suffer and to work, taught by an instinct, that so shall the dumb abyss be vocal with speech.”
—Ralph Waldo Emerson (1803–1882)