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Homediscrete valuation ring

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# 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 $t\in R$.

Defines:

uniformizer, uniformizing element, order

Related:

LocalRing, DiscreteValuation, Valuation

Synonym:

DVR

Type of Math Object:

Definition

Major Section:

Reference

## Mathematics Subject Classification

13F30*no label found*13H10

*no label found*

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