Localization Of A Ring

High Quality Content by WIKIPEDIA articles! 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 −1R; 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. The two latter cases are fundamental in algebraic theory and scheme theory. In particular the definition of an affine scheme is based on the properties of these two kinds of localizations. Dannoe izdanie predstavlyaet soboj kompilyatsiyu svedenij, nahodyaschihsya v svobodnom dostupe v srede Internet v tselom, i v informatsionnom setevom resurse "Vikipediya" v chastnosti. Sobrannaya po chastotnym zaprosam ukazannoj tematiki, dannaya kompilyatsiya postroena po printsipu podbora blizkih informatsionnyh ssylok, ne imeet samostoyatelnogo syuzheta, ne soderzhit nikakih analiticheskih materialov, vyvodov, otsenok moralnogo, eticheskogo, politicheskogo, religioznogo i mirovozzrencheskogo haraktera v otnoshenii glavnoj tematiki, predstavlyaya soboj isklyuchitelno faktologicheskij material. This book was created using print-on-demand technology.