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In mathematics, the projective line over a ring is an extension of the concept of projective line over a field. Given a ring ''A'' with 1, the projective line P(''A'') over ''A'' consists of points identified by homogeneous coordinates. Let ''U'' be the group of units of ''A''; pairs (''a'',''b'') and (''c'',''d'') from are related when there is a ''u'' in ''U'' such that and . This relation is an equivalence relation. A typical equivalence class is written ''U''(''a'',''b''). that is, ''U''(''a'',''b'') is in the projective line if the ideal generated by ''a'' and ''b'' is all of ''A''. The projective line P(''A'') is equipped with a group of homographies. The homographies are expressed through use of the matrix ring over ''A'' and its group of units ''V'' as follows: If ''c'' is in Z(''U''), the center of ''U'', then the group action of matrix on P(''A'') is the same as the action of the identity matrix. Such matrices represent a normal subgroup ''N'' of ''V''. The homographies of P(''A'') correspond to elements of the quotient group . P(''A'') is considered an extension of the ring ''A'' since it contains a copy of ''A'' due to the embedding . The multiplicative inverse mapping , ordinarily restricted to the group of units ''U'' of ''A'', is expressed by a homography on P(''A''): : Furthermore, for , the mapping can be extended to a homography: : : Since ''u'' is arbitrary, it may be substituted for ''u''−1. Homographies on P(''A'') are called linear-fractional transformations since :. ==Instances== Finite rings have finite projective lines. The projective line over GF(2) has three elements: ''U''(0,1), ''U''(1,0), and ''U''(1,1). Its homography group is the permutation group on these three.〔Robert Alexander Rankin (1977) ''Modular forms and functions'', page 29, Cambridge University Press ISBN 0-521-21212-X〕 The ring Z/3Z, or GF(3), has the elements 1, 0, and −1; its projective line has the four elements ''U''(1,0), ''U''(1,1), ''U''(0,1), ''U''(1,−1) since both 1 and −1 are units. The homography group on this projective line has 12 elements, also described with matrices or as permutations.〔Rankin page 31〕 The projective line over the ring of integers Z includes points ''U''(''m'',''n'') where ''n'' and ''m'' are relatively prime. The homography group on this projective line is the modular group. Its congruence subgroups serve as homography groups on projective lines over Z/''n''Z.〔A Blunck & H Havlicek (2000) "Projective representations: projective lines over rings", Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 70:287–99, . This article uses an alternative definition of projective line over a ring that restricts elements of the projective line over Z to those of the form ''U''(''m'',''n'') where ''m'' and ''n'' are coprime.〕〔Metod Saniga, Michel Planat, Maurice R. Kibler, Petr Pracna (2007) "A classification of the projective lines over small rings", Chaos, Solitons & Fractals 33(4):1095–1102, 〕 The projective line over a division ring results in a single auxiliary point . Examples include the real projective line, the complex projective line, and the projective line over quaternions. These examples of topological rings have the projective line as their one-point compactifications. The case of the complex number field has the Möbius group as its homography group. The projective line over the dual numbers was described by Josef Grünwald in 1906.〔Josef Grünwald (1906) "Über duale Zahlen und ihre Anwendung in der Geometrie", ''Monatshefte für Mathematik'' 17: 81–136〕 This ring includes a nonzero nilpotent ''n'' satisfying . The plane of dual numbers has a projective line including a line of points .〔Corrado Segre (1912) "Le geometrie proiettive nei campi di numeri duali", Paper XL of ''Opere'', also ''Atti della R. Academia della Scienze di Torino'', vol XLVII. 〕 Isaak Yaglom has described it as an "inversive Galilean plane" that has the topology of a cylinder when the supplementary line is included.〔I. M. Yaglom (1979) ''A Simple Non-Euclidean Geometry and its Physical Basis'', pp 149–53, Springer, ISBN 0387-90332-1, 〕 Similarly, if ''A'' is a local ring, then P(''A'') is formed by adjoining points corresponding to the elements of the maximal ideal of ''A''. The projective line over the ring ''M'' of split-complex numbers introduces auxiliary lines and Using stereographic projection the plane of split-complex numbers is closed up with these lines to a hyperboloid of one sheet.〔Yaglom 1979 p. 174–200〕〔Benz 1973〕 The projective line over ''M'' may be called the Minkowski plane when characterized by behaviour of hyperbolas under homographic mapping. 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「projective line over a ring」の詳細全文を読む スポンサード リンク
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