Editing Projective line over a ring (section)

=== Over continuous rings ===
The projective line over a [[division ring]] results in a single auxiliary point {{nowrap|1=∞ = ''U''[1, 0]}}. Examples include the [[real projective line]], the [[complex projective line]], and the projective line over [[quaternion]]s. These examples of [[topological ring]]s have the projective line as their [[one-point compactification]]s. The case of the [[complex number]] field '''C''' has the [[Möbius group]] as its homography group.

The projective line over the [[dual number]]s was described by Josef Grünwald in 1906.<ref name="Grünwald">{{citation |first=Josef |last=Grünwald |date=1906 |title=Über duale Zahlen und ihre Anwendung in der Geometrie |journal=[[Monatshefte für Mathematik]] |volume=17 |pages=81–136 |doi=10.1007/BF01697639 }}</ref> This ring includes a nonzero [[nilpotent]] ''n'' satisfying {{nowrap|1=''nn'' = 0}}. The plane {{nowrap|{{mset| ''z'' {{=}} ''x'' + ''yn'' | ''x'', ''y'' ∈ '''R''' }}}} of dual numbers has a projective line including a line of points {{nowrap|''U''[1, ''xn''], ''x'' ∈ '''R'''}}.<ref name=CS>[[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.</ref> [[Isaak Yaglom]] has described it as an "inversive Galilean plane" that has the [[topology]] of a [[cylinder (geometry)|cylinder]] when the supplementary line is included.<ref name=Yaglom79>{{citation |last=Yaglom |first=Isaak |authorlink=Isaak Yaglom |date=1979 |title=A Simple Non-Euclidean Geometry and its Physical Basis |publisher=Springer |isbn=0387-90332-1 |mr=520230 }}</ref>{{rp|149–153}} Similarly, if ''A'' is a [[local ring]], then P<sup>1</sup>(''A'') is formed  by adjoining points corresponding to the elements of the [[maximal ideal]] of&nbsp;''A''.

The projective line over the ring ''M'' of [[split-complex number]]s introduces auxiliary lines {{nowrap|{{mset| ''U''[1, ''x''(1 + j)] | ''x'' ∈ '''R''' }}}} and {{nowrap|{{mset| ''U''[1, ''x''(1 − j)] | ''x'' ∈ '''R''' }}}} Using [[stereographic projection]] the plane of split-complex numbers is [[motor variable#Compactification|closed up]] with these lines to a [[hyperboloid]] of one sheet.<ref name=Yaglom79/>{{rp| 174–200}}<ref name=Benz73>[[Walter Benz]] (1973) ''Vorlesungen über Geometrie der Algebren'', §2.1 Projective Gerade über einem Ring, §2.1.2 Die projective Gruppe, §2.1.3 Transitivitätseigenschaften, §2.1.4 Doppelverhaltnisse, Springer {{isbn|0-387-05786-2}} {{MathSciNet|id=353137}}</ref> The projective line over ''M'' may be called the [[Minkowski plane]] when characterized by behaviour of hyperbolas under homographic mapping.