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:

  1. R is a local principal ideal domain, and not a field.
  2. R is a valuation ring with a value group isomorphic to the integers under addition.
  3. R is a local Dedekind domain and not a field.
  4. R is a Noetherian local ring with Krull dimension one, and the maximal ideal of R is principal.
  5. R is an integrally closed Noetherian local ring with Krull dimension one.
  6. R is a principal ideal domain with a unique non-zero prime ideal.
  7. R is a principal ideal domain with a unique irreducible element (up to multiplication by units).
  8. R is a unique factorization domain with a unique irreducible element (up to multiplication by units).
  9. 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.
  10. 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) := { z ⁄ n : z, n ∈ Z, n 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 2k z ⁄ n, where z, n, and k are integers with z and n 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 \mathbb Z_{(p)}:=\left.\left\{\frac zn\,\right| z,n\in\mathbb Z,p\nmid n\right\} 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 Zp 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 pk 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 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.

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:

|x-y| = 2^{-\nu(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. 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 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 \Z_{(p)}=\Q \cap \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 Zp.

See also

References

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