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In abstract algebra, localization is a systematic method of adding multiplicative inverses to a ring. Given a ring ''R'' and a subset ''S'', one wants to construct some ring ''R *'' and ring homomorphism from ''R'' to ''R *'', such that the image of ''S'' consists of ''units'' (invertible elements) in ''R *''. Further one wants ''R *'' to be the 'best possible' or 'most general' way to do this – in the usual fashion this should be expressed by a universal property. The localization of ''R'' by ''S'' is usually denoted by ''S'' −1''R''; however other notations are used in some important special cases. If ''S'' is the set of the non zero elements of an integral domain, then the localization is the field of fractions and thus usually denoted Frac(''R''). If ''S'' is the complement of a prime ideal ''I'' the localization is denoted by ''RI'', and ''Rf'' is used to denote the localization by the powers of an element ''f''.〔Eisenbud, Harris, ''The geometry of schemes''〕 The two latter cases are fundamental in algebraic geometry and scheme theory. In particular the definition of an affine scheme is based on the properties of these two kinds of localizations. An important related process is completion: one often localizes a ring, then completes. == Terminology == The term ''localization'' originates in algebraic geometry: if ''R'' is a ring of functions defined on some geometric object (algebraic variety) ''V'', and one wants to study this variety "locally" near a point ''p'', then one considers the set ''S'' of all functions that are not zero at ''p'' and localizes ''R'' with respect to ''S''. The resulting ring ''R *'' contains only information about the behavior of ''V'' near ''p''. Cf. the example given at local ring. In number theory and algebraic topology, one refers to the behavior of a ring ''at'' a number ''n'' or ''away'' from ''n''. "Away from ''n''" means "in the ring localized by the set of the powers of ''n''" (which is a Z()-algebra). If ''n'' is a prime number, "at ''n''" means "in the ring localized by the set of the integers which are not multiple of ''n''". 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「localization of a ring」の詳細全文を読む スポンサード リンク
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