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

### Other articles related to "valuation ring, ring, valuation":

**Valuation Ring**

... 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 ...

**Valuation Ring**- Principal Ideal Domains

... 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 (1914–1953)