# 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