Discrete Valuation Ring - Examples

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 2kp/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 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] 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 pk divides x.