# Discrete valuation ring

In abstract algebra, a **discrete valuation ring** (**DVR**) is a principal ideal domain (PID) with exactly one non-zero maximal ideal.

This means a DVR is an integral domain *R* which satisfies any one of the following equivalent conditions:

*R*is a local principal ideal domain, and not a field.*R*is a valuation ring with a value group isomorphic to the integers under addition.*R*is a local Dedekind domain and not a field.*R*is a noetherian local ring with Krull dimension one, and the maximal ideal of*R*is principal.*R*is an integrally closed noetherian local ring with Krull dimension one.*R*is a principal ideal domain with a unique non-zero prime ideal.*R*is a principal ideal domain with a unique irreducible element (up to multiplication by units).*R*is a unique factorization domain with a unique irreducible element (up to multiplication by units).*R*is not a field, and every nonzero fractional ideal of*R*is irreducible in the sense that it cannot be written as finite intersection of fractional ideals properly containing it.- There is some discrete valuation ν on the field of fractions
*K*of*R*, such that*R*={*x*:*x*in*K*, ν(*x*) ≥ 0}.

## Examples

Let **Z**_{(2)}={ *p*/*q* : *p*, *q* in **Z**, *q* odd }. Then the field of fractions of **Z**_{(2)} is **Q**. Now, for any nonzero element *r* of **Q**, we can apply unique factorization to the numerator and denominator of *r* to write *r* as 2^{k}*p*/*q*, where *p*, *q*, and *k* are integers with *p* and *q* odd. In this case, we define ν(*r*)=*k*.
Then **Z**_{(2)} is the discrete valuation ring corresponding to ν. The maximal ideal of **Z**_{(2)} is the principal ideal generated by 2, and the "unique" irreducible element (up to units) is 2.

Note that **Z**_{(2)} is the localization of the Dedekind domain **Z** at the prime ideal generated by 2. Any localization of a Dedekind domain at a non-zero prime ideal is a discrete valuation ring; in practice, this is frequently how discrete valuation rings arise. In particular, we can define rings **Z**_{(p)} for any prime *p* in complete analogy.

For an example more geometrical in nature, take the ring *R* = {*f*/*g* : *f*, *g* polynomials in **R**[*X*] and *g*(0) ≠ 0}, considered as a subring of the field of rational functions **R**(*X*) in the variable *X*. *R* can be identified with the ring of all real-valued rational functions defined (i.e. finite) in a neighborhood of 0 on the real axis (with the neighborhood depending on the function). It is a discrete valuation ring; the "unique" irreducible element is *X* and the valuation assigns to each function *f* the order (possibly 0) of the zero of *f* at 0. This example provides the template for studying general algebraic curves near non-singular points, the algebraic curve in this case being the real line.

Another important example of a DVR is the ring of formal power series *R* = *K*[[T]] in one variable *T* over some field *K*. The "unique" irreducible element is *T*, the maximal ideal of *R* is the principal ideal generated by *T*, and the valuation ν assigns to each power series the index (i.e. degree) of the first non-zero coefficient.

If we restrict ourselves to real or complex coefficients, we can consider the ring of power series in one variable that *converge* in a neighborhood of 0 (with the neighborhood depending on the power series). This is also a discrete valuation ring.

Finally, the ring **Z**_{p} of *p*-adic integers is a DVR, for any prime *p*. Here *p* is an irreducible element; the valuation assigns to each *p*-adic integer *x* the largest integer *k* such that *p*^{k} divides *x*.

## Uniformizing parameter

Given a DVR *R*, then any irreducible element of *R* is a generator for the unique maximal ideal of *R* and vice versa. Such an element is also called a **uniformizing parameter** of *R* (or a **uniformizing element**, a **uniformizer**, or a **prime element**).

If we fix a uniformizing parameter *t*, then *M*=(*t*) is the unique maximal ideal of *R*, and every other non-zero ideal is a power of *M*, i.e. has the form (*t*^{ k}) for some *k*≥0. All the powers of *t* are distinct, and so are the powers of *M*. Every non-zero element *x* of *R* can be written in the form α*t*^{ k} with α a unit in *R* and *k*≥0, both uniquely determined by *x*. The valuation is given by *ν*(*x*) = *k*. So to understand the ring completely, one needs to know the group of units of *R* and how the units interact additively with the powers of *t*.

The function *v* also makes any discrete valuation ring into a Euclidean domain.

## Topology

Every discrete valuation ring, being a local ring, carries a natural topology and is a topological ring. The distance between two elements *x* and *y* can be measured as follows:

(or with any other fixed real number > 1 in place of 2). Intuitively: an element *z* is "small" and "close to 0" iff its valuation ν(*z*) is large. The function |x-y|, supplemented by |0|=0, is the restriction of an absolute value defined on the field
of fractions of the discrete valuation ring.

A DVR is compact if and only if it is complete and its residue field *R*/*M* is a finite field.

Examples of complete DVRs include the ring of *p*-adic integers and the ring of formal power series over any field. For a given DVR, one often passes to its completion, a complete DVR containing the given ring that is often easier to study. This completion procedure can be thought of in a geometrical way as passing from rational functions to power series, or rational numbers to the reals.

Returning to our examples: the ring of all formal power series in one variable with real coefficients is the completion of the ring of rational functions defined (i.e. finite) in a neighborhood of 0 on the real line; it is also the completion of the ring of all real power series that converge near 0. The completion of **Z**_{(p)} (which can be seen as the set of all rational numbers that are *p*-adic integers) is the ring of all *p*-adic integers **Z**_{p}.

## See also

## References

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