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:
Examples.
Algebraic.
Localization of Dedekind rings.
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 ringsformula_3for any prime "p" in complete analogy.
p-adic integers.
The ring formula_4 of "p"-adic integers is a DVR, for any prime formula_5. Here formula_5 is an irreducible element; the valuation assigns to each formula_5-adic integer formula_8 the largest integer formula_9 such that formula_10 divides formula_8.
Localisation of formula_12 at formula_13.
Let formula_14. Then, the field of fractions of formula_15 is formula_16. For any nonzero element formula_17 of formula_16, we can apply unique factorization to the numerator and denominator of "r" to write "r" as where "z", "n", and "k" are integers with "z" and "n" odd. In this case, we define ν("r")="k".
Then formula_15 is the discrete valuation ring corresponding to ν. The maximal ideal of formula_15 is the principal ideal generated by 2, i.e. formula_21, and the "unique" irreducible element (up to units) is 2 (this is also known as a uniformizing parameter).
Note that formula_15 is the localization of the Dedekind domain formula_12 at the prime ideal generated by 2.
Formal power series.
Another important example of a DVR is the ring of formal power series formula_24 in one variable formula_25 over some field formula_9. The "unique" irreducible element is formula_25, the maximal ideal of formula_28 is the principal ideal generated by formula_25, and the valuation formula_30 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 a discrete valuation ring. This is useful for building intuition with the Valuative criterion of properness.
Ring in function field.
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.
Scheme-Theoretic.
Henselian trait.
For a DVR formula_28 it is common to write the fraction field as formula_32 and formula_33 the residue field. These correspond to the generic and closed points of formula_34 For example, the closed point of formula_35 is formula_36 and the generic point is formula_37. Sometimes this is denoted as
where formula_39 is the generic point and formula_40 is the closed point .
Localization of a point on a curve.
Given an algebraic curve formula_41, the local ring formula_42 at a smooth point formula_43 is a discrete valuation ring, because it is a principal valuation ring. Note because the point formula_43 is smooth, the completion of the local ring is isomorphic to the completion of the localization of formula_45 at some point formula_46.
Uniformizing parameter.
Given a DVR "R", 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") = "kv"("t"). 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. We can also give it a metric space structure where 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
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 from 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 formula_48 (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".