Inmathematics, theprojective line over a ring is an extension of the concept ofprojective line over afield. Given aringA (with 1), the projective line P¹(A) overA consists of points identified byprojective coordinates. LetA^× be thegroup of units ofA; pairs(a,b) and(c,d) fromA ×A are related when there is au inA^× such thatua =c andub =d. This relation is anequivalence relation. A typicalequivalence class is writtenU[a,b].

P¹(A) = {U[a,b] |aA +bA =A }, that is,U[a,b] is in the projective line if theone-sided ideal generated bya andb is all of A.

The projective line P¹(A) is equipped with agroup of homographies. The homographies are expressed through use of thematrix ring overA and its group of unitsV as follows: Ifc is in Z(A^×), thecenter ofA^×, then thegroup action of matrix on P¹(A) is the same as the action of the identity matrix. Such matrices represent anormal subgroupN ofV. The homographies of P¹(A) correspond to elements of thequotient groupV / N.

P¹(A) is considered an extension of the ringA since it contains a copy ofA due to the embeddingE :a →U[a, 1]. Themultiplicative inverse mappingu → 1/u, ordinarily restricted toA^×, is expressed by a homography on P¹(A):

Furthermore, foru,v ∈A^×, the mappinga →uav can be extended to a homography:

Sinceu is arbitrary, it may be substituted foru⁻¹.Homographies on P¹(A) are calledlinear-fractional transformations since

Rings that arefields are most familiar: The projective line overGF(2) has three elements:U[0, 1],U[1, 0], andU[1, 1]. Its homography group is thepermutation group on these three.^[1]: 29

The ringZ / 3Z, or GF(3), has the elements 1, 0, and −1; its projective line has the four elementsU[1, 0],U[1, 1],U[0, 1],U[1, −1] since both 1 and −1 areunits. The homography group on this projective line has 12 elements, also described with matrices or as permutations.^[1]: 31 For afinite field GF(q), the projective line is theGalois geometryPG(1,q).J. W. P. Hirschfeld has described theharmonic tetrads in the projective lines forq = 4, 5, 7, 8, 9.^[2]

ConsiderP¹(Z / nZ) whenn is acomposite number. Ifp andq are distinct primes dividingn, then⟨p⟩ and⟨q⟩ aremaximal ideals inZ / nZ and byBézout's identity there area andb inZ such thatap +bq =1, so thatU[p,q] is inP¹(Z / nZ) but it is not an image of an element under the canonical embedding. The whole ofP¹(Z / nZ) is filled out by elementsU[up,vq], whereu ≠v andu,v ∈A^×,A^× being the units ofZ / nZ. The instancesZ / nZ are given here forn = 6, 10, and 12, where according tomodular arithmetic the group of units of the ring is(Z / 6Z)^× = {1, 5},(Z / 10Z)^× = {1, 3, 7, 9}, and(Z / 12Z)^× = {1, 5, 7, 11} respectively. Modular arithmetic will confirm that, in each table, a given letter represents multiple points. In these tables a pointU[m,n] is labeled bym in the row at the table bottom andn in the column at the left of the table. For instance, thepoint at infinityA =U[v, 0], wherev is a unit of the ring.

The extra points can be associated withQ ⊂R ⊂C, the rationals in theextended complex upper-half plane. The group of homographies onP¹(Z / nZ) is called aprincipal congruence subgroup.^[3]

For therational numbersQ, homogeneity of coordinates means that every element of P¹(Q) may be represented by an element of P¹(Z). Similarly, a homography of P¹(Q) corresponds to an element of themodular group, the automorphisms of P¹(Z).

The projective line over adivision ring results in a single auxiliary point∞ =U[1, 0]. Examples include thereal projective line, thecomplex projective line, and the projective line overquaternions. These examples oftopological rings have the projective line as theirone-point compactifications. The case of thecomplex number fieldC has theMöbius group as its homography group.

The projective line over thedual numbers was described by Josef Grünwald in 1906.^[4] This ring includes a nonzeronilpotentn satisfyingnn = 0. The plane{z =x +yn |x,y ∈R } of dual numbers has a projective line including a line of pointsU[1,xn],x ∈R.^[5]Isaak Yaglom has described it as an "inversive Galilean plane" that has thetopology of acylinder when the supplementary line is included.^{[6]: 149–153} Similarly, ifA is alocal ring, then P¹(A) is formed by adjoining points corresponding to the elements of themaximal ideal of A.

The projective line over the ringM ofsplit-complex numbers introduces auxiliary lines{U[1,x(1 + j)] |x ∈R } and{U[1,x(1 − j)] |x ∈R } Usingstereographic projection the plane of split-complex numbers isclosed up with these lines to ahyperboloid of one sheet.^{[6]: 174–200 [7]} The projective line overM may be called theMinkowski plane when characterized by behaviour of hyperbolas under homographic mapping.

The projective line P¹(A) over a ringA can also be identified as the space ofprojective modules in themoduleA ⊕A. An element of P¹(A) is then adirect summand ofA ⊕A. This more abstract approach follows the view ofprojective geometry as the geometry ofsubspaces of avector space, sometimes associated with thelattice theory ofGarrett Birkhoff^[8] or the bookLinear Algebra and Projective Geometry byReinhold Baer. In the case of the ring of rationalintegersZ, the module summand definition of P¹(Z) narrows attention to theU[m,n],mcoprime ton, and sheds the embeddings that are a principal feature of P¹(A) whenA is topological. The 1981 article by W. Benz, Hans-Joachim Samaga, & Helmut Scheaffer mentions the direct summand definition.

In an article "Projective representations: projective lines over rings"^[9] thegroup of units of amatrix ring M₂(R) and the concepts of module andbimodule are used to define a projective line over a ring. The group of units is denoted byGL(2,R), adopting notation from thegeneral linear group, whereR is usually taken to be a field.

The projective line is the set of orbits underGL(2,R) of the free cyclicsubmoduleR(1, 0) ofR ×R. Extending the commutative theory of Benz, the existence of a right or leftmultiplicative inverse of a ring element is related to P¹(R) andGL(2,R). TheDedekind-finite property is characterized. Most significantly,representation of P¹(R) in a projective space over a division ringK is accomplished with a(K,R)-bimoduleU that is a leftK-vector space and a rightR-module. The points of P¹(R) are subspaces ofP¹(K,U ×U) isomorphic to their complements.

A homographyh that takes three particular ring elementsa,b,c to the projective line pointsU[0, 1],U[1, 1],U[1, 0] is called thecross-ratio homography. Sometimes^[10][11] thecross-ratio is taken as the value ofh on a fourth pointx : (x,a,b,c) =h(x).

To buildh froma,b,c the generator homographies

are used, with attention tofixed points: +1 and −1 are fixed under inversion,U[1, 0] is fixed under translation, and the "rotation" withu leavesU[0, 1] andU[1, 0] fixed. The instructions are to placec first, then bringa toU[0, 1] with translation, and finally to use rotation to moveb toU[1, 1].

Lemma: IfA is acommutative ring andb −a,c −b,c −a are all units, then(b −c)⁻¹ + (c −a)⁻¹ is a unit.

Proof: Evidently${\displaystyle \frac{b-a}{(b-c)(c-a)}=\frac{(b-c)+(c-a)}{(b-c)(c-a)}}$ is a unit, as required.

Theorem: If(b −c)⁻¹ + (c −a)⁻¹ is a unit, then there is a homographyh in G(A) such that

h(a) =U[0, 1],h(b) =U[1, 1], andh(c) =U[1, 0].

Proof: The pointp = (b −c)⁻¹ + (c −a)⁻¹ is the image ofb aftera was put to 0 and then inverted toU[1, 0], and the image ofc is brought toU[0, 1]. Asp is a unit, its inverse used in a rotation will movep toU[1, 1], resulting ina,b,c being all properly placed. The lemma refers to sufficient conditions for the existence ofh.

One application of cross ratio defines theprojective harmonic conjugate of a triplea,b,c, as the elementx satisfying(x,a,b,c) = −1. Such a quadruple is aharmonic tetrad. Harmonic tetrads on the projective line over afinite field GF(q) were used in 1954 to delimit the projective linear groupsPGL(2,q) forq = 5, 7, and 9, and demonstrateaccidental isomorphisms.^[12]

Thereal line in thecomplex plane gets permuted with circles and other real lines underMöbius transformations, which actually permute the canonical embedding of thereal projective line in thecomplex projective line. SupposeA is analgebra over a fieldF, generalizing the case whereF is the real number field andA is the field of complex numbers. The canonical embedding of P¹(F) into P¹(A) is

${\displaystyle U_F[x,1]\mapsto U_A[x,1],U_F[1,0]\mapsto U_A[1,0].}$

Achain is the image of P¹(F) under a homography on P¹(A). Four points lie on a chainif and only if their cross-ratio is inF.Karl von Staudt exploited this property in his theory of "real strokes" [reeler Zug].^[13]

Two points of P¹(A) areparallel if there isno chain connecting them. The convention has been adopted that points are parallel to themselves. This relation isinvariant under the action of a homography on the projective line. Given three pair-wise non-parallel points, there is a unique chain that connects the three.^[14]

August Ferdinand Möbius investigated theMöbius transformations between his bookBarycentric Calculus (1827) and his 1855 paper "Theorie der Kreisverwandtschaft in rein geometrischer Darstellung".Karl Wilhelm Feuerbach andJulius Plücker are also credited with originating the use of homogeneous coordinates.Eduard Study in 1898, andÉlie Cartan in 1908, wrote articles onhypercomplex numbers for German and FrenchEncyclopedias of Mathematics, respectively, where they use these arithmetics withlinear fractional transformations in imitation of those of Möbius. In 1902Theodore Vahlen contributed a short but well-referenced paper exploring some linear fractional transformations of aClifford algebra.^[15] The ring ofdual numbersD gave Josef Grünwald opportunity to exhibit P¹(D) in 1906.^[4]Corrado Segre (1912) continued the development with that ring.^[5]

Arthur Conway, one of the early adopters of relativity viabiquaternion transformations, considered the quaternion-multiplicative-inverse transformation in his 1911 relativity study.^[16] In 1947 some elements of inversive quaternion geometry were described by P.G. Gormley in Ireland.^[17] In 1968Isaak Yaglom'sComplex Numbers in Geometry appeared in English, translated from Russian. There he uses P¹(D) to describeline geometry in the Euclidean plane and P¹(M) to describe it for Lobachevski's plane. Yaglom's textA Simple Non-Euclidean Geometry appeared in English in 1979. There in pages 174 to 200 he developsMinkowskian geometry and describes P¹(M) as the "inversive Minkowski plane". The Russian original of Yaglom's text was published in 1969. Between the two editions,Walter Benz (1973) published his book,^[7] which included the homogeneous coordinates taken from M.

• Euclid's orchard

• Mitod Saniga (2006)Projective Lines over Finite Rings (pdf) fromAstronomical Institute of the Slovak Academy of Sciences

• Algebraic geometry
• Ring theory
• Projective geometry

• Articles with short description
• Short description matches Wikidata
• CS1 maint: DOI inactive as of November 2024