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In abstract algebra, a valuation ring is an integral domain ''D'' such that for every element ''x'' of its field of fractions ''F'', at least one of ''x'' or ''x'' −1 belongs to ''D''. Given a field ''F'', if ''D'' is a subring of ''F'' such that either ''x'' or ''x'' −1 belongs to ''D'' for every nonzero ''x'' in ''F'', then ''D'' is said to be a valuation ring for the field ''F'' or a place of ''F''. Since ''F'' in this case is indeed the field of fractions of ''D'', a valuation ring for a field is a valuation ring. Another way to characterize the valuation rings of a field ''F'' is that valuation rings ''D'' of ''F'' have ''F'' as their field of fractions, and their ideals are totally ordered by inclusion; or equivalently their principal ideals are totally ordered by inclusion. In particular, every valuation ring is a local ring. The valuation rings of a field are the maximal elements of the set of the local subrings in the field partially ordered by dominance or refinement, where : dominates if and .〔Efrat (2006) p.55〕 Every local ring in a field ''K'' is dominated by some valuation ring of ''K''. An integral domain whose localization at any prime ideal is a valuation ring is called a Prüfer domain. == Examples == * Any field is a valuation ring. * Z(''p''), the localization of the integers Z at the prime ideal (''p''), consisting of ratios where the numerator is any integer and the denominator is not divisible by ''p''. The field of fractions is the field of rational numbers Q. * The ring of meromorphic functions on the entire complex plane which have a Maclaurin series (Taylor series expansion at zero) is a valuation ring. The field of fractions are the functions meromorphic on the whole plane. If ''f'' does not have a Maclaurin series then 1/''f'' does. * Any ring of p-adic integers Z''p'' for a given prime ''p'' is a local ring, with field of fractions the ''p''-adic numbers Qp. The integral closure Z''p''cl of the ''p''-adic integers is also a local ring, with field of fractions Q''p''cl (the algebraic closure of ''p''-adic numbers). Both Z''p'' and Z''p''cl are valuation rings. * Let k be an ordered field. An element of k is called finite if it lies between two integers ''n''<''x''<''m''; otherwise it is called infinite. The set ''D'' of finite elements of k is a valuation ring. The set of elements ''x'' such that ''x'' ∈ ''D'' and ''x''−1∉''D'' is the set of infinitesimal elements; and an element ''x'' such that ''x''∉''D'' and ''x''−1∈''D'' is called infinite. * The ring F of finite elements of a hyperreal field *R (an ordered field containing the real numbers) is a valuation ring of *R. F consists of all hyperreal numbers differing from a standard real by an infinitesimal amount, which is equivalent to saying a hyperreal number ''x'' such that −''n'' < ''x'' < ''n'' for some standard integer ''n''. The residue field, finite hyperreal numbers modulo the ideal of infinitesimal hyperreal numbers, is isomorphic to the real numbers. 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「Valuation ring」の詳細全文を読む スポンサード リンク
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