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Formally, a DVR is an Integral Domain ''R'' which satisfies any one of the following equivalent conditions: # ''R'' is a Local Principal Ideal Domain , and not a Field . # ''R'' is a Noetherian Local Ring with positive Krull Dimension , and the maximal Ideal of ''R'' is principal. # ''R'' is a Local Dedekind Domain and not a field. # ''R'' is a Unique Factorization Domain with a unique Irreducible Element ( Up To multiplication with Unit s). # ''R'' is Local , not a Field , and every nonzero Fractional Ideal of ''R'' is irreducible. # There is some Dedekind Valuation ν on the Field Of Fractions ''K'' of ''R'', such that ''R''={''x'':''x'' in ''K'', ν(''x'') ≥ 0}. 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''k''''p''/''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 prinicipal 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'' Polynomial s in R {Link without Title} and ''g''(0) ≠ 0}, considered as a Subring of the field of Rational Function s 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 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'' 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''. UNIFORMIZING PARAMETER Given a DVR ''R'', then any irreducible element of ''R'' is as a generator for the unique maximal ideal of ''R'' and vice versa. Such an element is also called a uniformizing parameter or ''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 α''t'' ''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''. 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: |
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