alternative definition of Krull valuation

Let G be an abelian totally ordered group, denoted additively. We adjoin to G a new element such that g<+, for all gG and we extend the additionPlanetmathPlanetmath on G=G{+} by declaring g+(+)=(+)+(+)=+.

Definition 1.

Let R be an unital ring, a valuationMathworldPlanetmathPlanetmath of R with values in G is a function from R to G such that , for all x,yR:

1) v(xy)=v(x)+v(y),

2) v(x+y)min{v(x),v(y)},

3) v(x)=+ iff v(x)=0.

Remarks a) The condition 1) means that v is a homomorhism of R{0} with multiplication in the group G. In particular, v(1)=0 and v(-x)=v(x), for all xG. If x is invertiblePlanetmathPlanetmath then 0=v(1)=v(xx-1)=v(x)+v(x-1), so v(x-1)=-v(x).
b) If 3) is replaced by the condition v(0)=+ then the set P=v-1{+} is a prime idealMathworldPlanetmathPlanetmath of R and v is on the integral domainMathworldPlanetmath R/P.
c) In particular, conditions 1) and 3) that R is an integral domain and let K be its quotient field. There is a unique valuation of K with values in G that extends v, namely v(x/y)=v(x)-v(y), for all xR and yR{0}.
d) The element v(x) is sometimes denoted by vx.

Title alternative definition of Krull valuation
Canonical name AlternativeDefinitionOfKrullValuation
Date of creation 2013-03-22 17:02:08
Last modified on 2013-03-22 17:02:08
Owner polarbear (3475)
Last modified by polarbear (3475)
Numerical id 10
Author polarbear (3475)
Entry type Definition
Classification msc 13F30
Classification msc 13A18
Classification msc 12J20
Classification msc 11R99
Related topic OrderValuation
Related topic Valuation
Related topic Krullvaluation
Related topic KrullValuation