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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: # ''R'' is a local principal ideal domain, and not a field. # ''R'' is a valuation ring with a value group isomorphic to the integers under addition. # ''R'' is a local Dedekind domain and not a field. # ''R'' is a Noetherian local ring with Krull dimension one, and the maximal ideal of ''R'' is principal. # ''R'' is an integrally closed Noetherian local ring with Krull dimension one. # ''R'' is a principal ideal domain with a unique non-zero prime ideal. # ''R'' is a principal ideal domain with a unique irreducible element (up to multiplication by units). # ''R'' is a unique factorization domain with a unique irreducible element (up to multiplication by units). # ''R'' is not a field, and every nonzero fractional ideal of ''R'' is irreducible in the sense that it cannot be written as finite intersection of fractional ideals properly containing it. # There is some discrete valuation ν on the field of fractions ''K'' of ''R'', such that ''R''=. ==Examples== Let Z(2)=. 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 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'' = , 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. 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''. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Discrete valuation ring」の詳細全文を読む スポンサード リンク
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