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:
“This is the gospel of labour, ring it, ye bells of the kirk!
The Lord of Love came down from above, to live with the men who work.
This is the rose that He planted, here in the thorn-curst soil:
Heaven is blest with perfect rest, but the blessing of Earth is toil.”
—Henry Van Dyke (1852–1933)