Editing Projective line over a ring (section)

== Instances ==
[[File:Projectivisation_F5P%5E1.svg|thumb|200px|Six colors illustrate the projective line over Galois field GF(5)]]

[[File:Lines through origin finite field 25.jpg|thumb|Six lines through the origin in F_25, each corresponding to a point in the projective line P(F_5).]]

Rings that are [[field (mathematics)|field]]s are most familiar: The projective line over [[GF(2)]] has three elements: {{nowrap|''U''[0, 1]}}, {{nowrap|''U''[1, 0]}}, and {{nowrap|''U''[1, 1]}}. Its homography group is the [[permutation group]] on these three.<ref name=Rankin>{{citation |last=Rankin |first=R.A. |authorlink=Robert Alexander Rankin |date=1977 |title=Modular forms and functions |publisher=[[Cambridge University Press]] |isbn=0-521-21212-X }}</ref>{{rp|29}}

The ring [[Modular arithmetic#Integers modulo m|'''Z'''{{hsp}}/{{hsp}}3'''Z''']], or GF(3), has the elements 1, 0, and −1; its projective line has the four elements {{nowrap|''U''[1, 0]}}, {{nowrap|''U''[1, 1]}}, {{nowrap|''U''[0, 1]}}, {{nowrap|''U''[1, −1]}} since both 1 and −1 are [[unit (ring theory)|unit]]s. The homography group on this projective line has 12 elements, also described with matrices or as permutations.<ref name=Rankin/>{{rp|31}} For a [[finite field]] GF(''q''), the projective line is the [[Galois geometry]] {{nowrap|PG(1, ''q'')}}. [[J. W. P. Hirschfeld]] has described the [[projective harmonic conjugate#Galois tetrads|harmonic tetrads]] in the projective lines for ''q'' = 4, 5, 7, 8, 9.<ref>{{cite book |title=Projective Geometries Over Finite Fields |first1=J. W. P. |last1=Hirschfeld |authorlink=J. W. P. Hirschfeld |publisher=[[Oxford University Press]] |year=1979 |page=129 |isbn=978-0-19-850295-1 }}</ref>

=== Over discrete rings ===
Consider {{nowrap|P<sup>1</sup>('''Z'''{{hsp}}/{{hsp}}''n'''''Z''')}} when ''n'' is a [[composite number]]. If ''p'' and ''q'' are distinct primes dividing ''n'', then {{angle bracket|''p''}} and {{angle bracket|''q''}} are [[maximal ideal]]s in {{nowrap|'''Z'''{{hsp}}/{{hsp}}''n'''''Z'''}} and by [[Bézout's identity]] there are ''a'' and ''b'' in '''Z''' such that {{nowrap|1=''ap'' + ''bq'' = ''1''}}, so that {{nowrap|''U''[''p'', ''q'']}} is in {{nowrap|P<sup>1</sup>('''Z'''{{hsp}}/{{hsp}}''n'''''Z''')}} but it is not an image of an element under the canonical embedding. The whole of {{nowrap|P<sup>1</sup>('''Z'''{{hsp}}/{{hsp}}''n'''''Z''')}} is filled out by elements {{nowrap|''U''[''up'', ''vq'']}}, where {{nowrap|''u'' ≠ ''v''}} and {{nowrap|''u'', ''v'' ∈ ''A''<sup>×</sup>}}, ''A''<sup>×</sup> being the units of {{nowrap|'''Z'''{{hsp}}/{{hsp}}''n'''''Z'''}}. The instances {{nowrap|'''Z'''{{hsp}}/{{hsp}}''n'''''Z'''}} are given here for ''n'' = 6, 10, and 12, where according to [[modular arithmetic]] the group of units of the ring is {{nowrap|1=('''Z'''{{hsp}}/{{hsp}}6'''Z''')<sup>×</sup> = {{mset|1, 5}}}}, {{nowrap|1=('''Z'''{{hsp}}/{{hsp}}10'''Z''')<sup>×</sup> = {{mset|1, 3, 7, 9}}}}, and {{nowrap|1=('''Z'''{{hsp}}/{{hsp}}12'''Z''')<sup>×</sup> = {{mset|1, 5, 7, 11}}}} respectively. Modular arithmetic will confirm that, in each table, a given letter represents multiple points. In these tables a point {{nowrap|''U''[''m'', ''n'']}} is labeled by ''m'' in the row at the table bottom and ''n'' in the column at the left of the table. For instance, the [[point at infinity]] {{nowrap|1=A = ''U''[''v'', 0]}}, where ''v'' is a unit of the ring.

{|
|style="width: 20em;"|
{| class="wikitable" style="text-align: center;"
|+ Projective line over the ring {{nowrap|'''Z'''{{hsp}}/{{hsp}}6'''Z'''}}
 ! 5
 | B || G || F || E || D || C
|-
 ! 4
 |   || J ||   || K ||   || H
|-
 ! 3
 |   || I || L ||   || L || I
|-
 ! 2
 |   || H ||   || K ||   || J
|-
 ! 1
 | B || C || D || E || F || G
|-
 ! 0
 |   || A ||   ||   ||   || A
|-
 ! 
 ! 0 !! 1 !! 2 !! 3 !! 4 !! 5
|}
|style="width: 30em;"| 
{| class="wikitable" style="text-align: center;"
|+ Projective line over the ring {{nowrap|'''Z'''{{hsp}}/{{hsp}}10'''Z'''}}
 ! 9
 | B || K || J || I || H || G || F || E || D || C
|-
 ! 8
 |   || P ||   || O ||   || Q ||   || M ||   || L
|-
 ! 7
 | B || E || H || K || D || G || J || C || F || I
|-
 ! 6
 |   || O ||   || L ||   || Q ||   || P ||   || M
|-
 ! 5
 |   || N || R || N || R ||   || R || N || R || N
|-
 ! 4
 |   || M ||   || P ||   || Q ||   || L ||   || O
|-
 ! 3
 | B || I || F || C || J || G || D || K || H || E
|-
 ! 2
 |   || L ||   || M ||   || Q ||   || O ||   || P
|-
 ! 1
 | B || C || D || E || F || G || H || I || J || K
|-
 ! 0
 |   || A ||   || A ||   ||   ||   || A ||   || A
|-
 ! 
 ! 0 !! 1 !! 2 !! 3 !! 4 !! 5 !! 6 !! 7 !! 8 !! 9
|}
| 
{| class="wikitable" style="text-align: center;"
|+ Projective line over the ring {{nowrap|'''Z'''{{hsp}}/{{hsp}}12'''Z'''}}
 ! 11
 | B || M || L || K || J || I || H || G || F || E || D || C
|-
 ! 10
 |   || T ||   || U ||   || N ||   || T ||   || U ||   || N
|-
 ! 9
 |   || S || V ||   || W || S ||   || O || W ||   || V || O
|-
 ! 8
 |   || R ||   || X ||   || P ||   || R ||   || X ||   || P
|-
 ! 7
 | B || I || D || K || F || M || H || C || J || E || L || G
|-
 ! 6
 |   || Q ||   ||   ||   || Q ||   || Q ||   ||   ||   || Q
|-
 ! 5
 | B || G || L || E || J || C || H || M || F || K || D || I
|-
 ! 4
 |   || P ||   || X ||   || R ||   || P ||   || X ||   || R
|-
 ! 3
 |   || O || V ||   || W || O ||   || S || W ||   || V || S
|-
 ! 2
 |   || N ||   || U ||   || T ||   || N ||   || U ||   || T
|-
 ! 1
 | B || C || D || E || F || G || H || I || J || K || L || M
|-
 ! 0
 |   || A ||   ||   ||   || A ||   || A ||   ||   ||   || A
|-
 !
 ! 0 !! 1 !! 2 !! 3 !! 4 !! 5 !! 6 !! 7 !! 8 !! 9 !!10 !!11
|}
|+ Tables showing the projective lines over rings {{nowrap|'''Z'''{{hsp}}/{{hsp}}''n'''''Z'''}} for ''n'' = 6, 10, 12. Ordered pairs marked with the same letter belong to the same point.
|}

The extra points can be associated with {{nowrap|'''Q''' ⊂ '''R''' ⊂ '''C'''}}, the rationals in the [[extended complex upper-half plane]]. The group of homographies on {{nowrap|P<sup>1</sup>('''Z'''{{hsp}}/{{hsp}}''n'''''Z''')}} is called a [[congruence subgroup|principal congruence subgroup]].<ref>{{citation |first1=Metod |last1=Saniga |first2=Michel |last2=Planat |first3=Maurice R. |last3=Kibler |first4=Petr |last4=Pracna |date=2007 |title=A classification of the projective lines over small rings |journal=[[Chaos, Solitons & Fractals]] |volume=33 |issue=4 |pages=1095–1102 |doi=10.1016/j.chaos.2007.01.008 |arxiv=math/0605301 |bibcode=2007CSF....33.1095S |mr=2318902 }}</ref>

For the [[rational number]]s '''Q''', homogeneity of coordinates means that every element of P<sup>1</sup>('''Q''') may be represented by an element of P<sup>1</sup>('''Z'''). Similarly, a homography of P<sup>1</sup>('''Q''') corresponds to an element of the [[modular group]], the automorphisms of P<sup>1</sup>('''Z''').

=== 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.