discrete valuation ring

A discrete valuation ring R is a principal ideal domainMathworldPlanetmath with exactly one nonzero maximal idealMathworldPlanetmath M. Any generator t of M is called a uniformizer or uniformizing element of R; in other words, a uniformizer of R is an element tR such that tM but tM2.

Given a discrete valuation ring R and a uniformizer tR, every element zR can be written uniquely in the form utn for some unit uR and some nonnegative integer n. The integer n is called the order of z, and its value is independent of the choice of uniformizing element tR.

Title discrete valuation ring
Canonical name DiscreteValuationRing
Date of creation 2013-03-22 12:16:40
Last modified on 2013-03-22 12:16:40
Owner djao (24)
Last modified by djao (24)
Numerical id 9
Author djao (24)
Entry type Definition
Classification msc 13F30
Classification msc 13H10
Synonym DVR
Related topic LocalRing
Related topic DiscreteValuation
Related topic ValuationMathworldPlanetmath
Defines uniformizer
Defines uniformizing element
Defines order