Discrete valuation ring

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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 unique factorization domain with a unique irreducible element (up to multiplication with units).
7. R is local, not a field, and every nonzero fractional ideal of R is irreducible.
8. 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 1 Examples 2 Uniformizing parameter 3 Topology 4 References

[edit] 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^kp/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 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 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.

[edit] 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.

If we fix a uniformizing parameter, 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 different, and so are the powers of M. Every 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.

[edit] 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 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.

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 p-adic integers) is the ring of all p-adic integers Z_p.

[edit] References

• Atiyah, M. and I.G. MacDonald Introduction to Commutative Algebra (1994)
• Dummit, David and Richard Foote, Abstract Algebra (2003).
• Discrete valuation ring, The Encyclopaedia of Mathematics.