Discrete valuation ring

In abstract algebra, a discrete valuation ring (DVR) is a principal ideal domain (PID) with exactly one nonzero 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 nonzero 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 Dedekind valuation ν on the field of fractions K of R, such that R={x : x in K, ν(x) ≥ 0}.
Contents
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 nonzero 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 realvalued rational functions defined 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 nonsingular 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 of the first nonzero 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 padic integers is a DVR, for any prime p. Here p is an irreducible element; the valuation assigns to each padic 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, or a uniformizer).
If we fix a uniformizing parameter, then M=(t) is the unique maximal ideal of R, and every other nonzero ideal is a power of M, i.e. has the form (t^{ k}) for some k≥0. All the powers of t are different, and so are the powers of M. Every nonzero element x of R can be written in the form αt^{ 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.
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:
  x − y  = 2 ^{− ν(x − y)}
(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.
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 padic 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.
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 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 padic integers) is the ring of all padic integers Z_{p}.
See also
References
 Atiyah, Michael Francis; Macdonald, I.G. (1969), Introduction to Commutative Algebra, Westview Press, ISBN 9780201407518
 Dummit, David S.; Foote, Richard M. (2004), Abstract algebra (3rd ed.), New York: John Wiley & Sons, ISBN 9780471433347, MR2286236
 Discrete valuation ring, The Encyclopaedia of Mathematics.
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