ring of formal power series in one variable over some field . The "unique" irreducible element is , the maximal ideal of is the principal ideal generated by , and the valuation 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 a discrete valuation ring. This is useful for building intuition with the Valuative criterion of properness. Ring in function field 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. Scheme-theoretic Henselian trait For a DVR it is common to write the fraction field as and the residue field. These correspond to the generic and closed points of For example, the closed point of is and the generic point is . Sometimes this is denoted as where is the generic point and is the closed point . Localization of a point on a curve Given an algebraic curve , the local ring at a smooth point is a discrete valuation ring, because it is a principal valuation ring. Note because the point is smooth, the completion of the local ring is isomorphic to the completion of the localization of at some point . Uniformizing parameter Given a DVR R, 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,