discrete valuation ring

A discrete valuation ring $R$ is a principal ideal domain with exactly one nonzero maximal ideal $M$. Any generator $t$ of $M$ is called a uniformizer or uniformizing element of $R$; in other words, a uniformizer of $R$ is an element $t\in R$ such that $t\in M$ but $t\notin M^2$.

Given a discrete valuation ring $R$ and a uniformizer $t\in R$, every element $z\in R$ can be written uniquely in the form $u\cdot t^n$ for some unit $u\in R$ and some nonnegative integer $n\in \mathbb{Z}$. The integer $n$ is called the order of $z$, and its value is independent of the choice of uniformizing element $t\in R$.

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	Valuation
Defines	uniformizer
Defines	uniformizing element
Defines	order