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 that satisfies any and all of the following equivalent conditions: Let, 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.is the discrete valuation ring corresponding to ν., and the "unique" irreducible element (up to units) is 2 (this is also known as a uniformizing parameter).More generally, 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 for any prime p in complete analogy.of p-adic integers is a DVR, for any primeis an irreducible element; the valuation assigns to eachAnother important example of a DVR is the ring of formal power seriesassigns 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 useful for building intuition with the Valuative criterion of properness.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.it is common to write the fraction field asThese correspond to the generic and closed points ofGiven a DVR R, any irreducible element of R is a generator for the unique maximal ideal of R and vice versa.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.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.It also admits 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.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.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.