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discrete valuation ring (Definition)

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 $t \in R$ .




"discrete valuation ring" is owned by djao. [ full author list (2) | owner history (1) ]
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See Also: local ring, discrete valuation, valuation

Other names:  DVR
Also defines:  uniformizer, uniformizing element, order

Attachments:
p-adic canonical form (Example) by pahio
ideals of a discrete valuation ring are powers of its maximal ideal (Theorem) by rm50
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Cross-references: independent, integer, unit, element, generator, maximal ideal, principal ideal domain
There are 95 references to this entry.

This is version 6 of discrete valuation ring, born on 2002-02-03, modified 2006-01-17.
Object id is 1727, canonical name is DiscreteValuationRing.
Accessed 19708 times total.

Classification:
AMS MSC13F30 (Commutative rings and algebras :: Arithmetic rings and other special rings :: Valuation rings)
 13H10 (Commutative rings and algebras :: Local rings and semilocal rings :: Special types )

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