Projective line over a ring

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 projective coordinates. Let U be the group of units of A; pairs (a, b) and (c, d) from A × A are related when there is a u in U such that ua = c and ub = d. This relation is an equivalence relation. A typical equivalence class is written U[a, b].

P(A) is considered an extension of the ring A since it contains a copy of A due to the embedding E : aU[a, 1]. The multiplicative inverse mapping u → 1/u, ordinarily restricted to the group of units U of A, is expressed by a homography on P(A):

Furthermore, for u,vU, the mapping auav 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

Rings that are fields are most familiar: 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.[1]: 29 

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.[1]: 31  For a finite field GF(q), the projective line is the Galois geometry PG(1, q). J. W. P. Hirschfeld has described the harmonic tetrads in the projective lines for q = 4, 5, 7, 8, 9.[2]

The extra points can be associated with QRC, the rationals in the extended complex upper-half plane. The group of homographies on P(Z/nZ) is called a principal congruence subgroup.[3]

The projective line over a division ring results in a single auxiliary point ∞ = U[1,0]. 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 C has the Möbius group as its homography group. For the rational numbers Q, homogeneity of coordinates means that every element of P(Q) may be represented by an element of P(Z). Similarly, a homography of P(Q) corresponds to an element of the modular group, the automorphisms of P(Z).

The real line in the complex plane gets permuted with circles and other real lines under Möbius transformations, which actually permute the canonical embedding of the real projective line in the complex projective line. Suppose A is an algebra over a field F, generalizing the case where F is the real number field and A is the field of complex numbers. The canonical embedding of P(F) into P(A) is

A chain is the image of P(F) under a homography on P(A). Four points lie on a chain if and only if their cross-ratio is in F. Karl von Staudt exploited this property in his theory of "real strokes" [reeler Zug].[8]

Two points of P(A) are parallel if there is no chain connecting them. The convention has been adopted that points are parallel to themselves. This relation is invariant 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.[9]

In an article "Projective representations: projective lines over rings"[11] the group of units of a matrix ring M2(R) and the concepts of module and bimodule are used to define a projective line over a ring. The group of units is denoted by GL(2,R), adopting notation from the general linear group, where R is usually taken to be a field.

The projective line is the set of orbits under GL(2,R) of the free cyclic submodule R(1,0) of R × R. Extending the commutative theory of Benz, the existence of a right or left multiplicative inverse of a ring element is related to P(R) and GL(2,R). The Dedekind-finite property is characterized. Most significantly, representation of P(R) in a projective space over a division ring K is accomplished with a (K,R)-bimodule U that is a left K-vector space and a right R-module. The points of P(R) are subspaces of P(K, U × U) isomorphic to their complements.

A homography h that takes three particular ring elements a, b, c to the projective line points U[0,1], U[1,1], U[1,0] is called the cross-ratio homography. Sometimes[12][13] the cross-ratio is taken as the value of h on a fourth point x : (x,a,b,c) = h(x).

are used, with attention to fixed points: +1 and −1 are fixed under inversion, U[1,0] is fixed under translation, and the "rotation" with u leaves U[0,1] and U[1,0] fixed. The instructions are to place c first, then bring a to U[0,1] with translation, and finally to use rotation to move b to U[1,1].

Lemma: If A is a commutative ring and ba, cb, ca are all units, then

One application of cross ratio defines the projective harmonic conjugate of a triple a, b, c, as the element x satisfying (x, a, b, c) = −1. Such a quadruple is a harmonic tetrad. Harmonic tetrads on the projective line over a finite field GF(q) were used in 1954 to delimit the projective linear groups PGL(2, q) for q = 5, 7, and 9, and demonstrate accidental isomorphisms.[14]

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

Arthur Conway, one of the early adopters of relativity via biquaternion 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 1968 Isaak Yaglom's Complex Numbers in Geometry appeared in English, translated from Russian. There he uses P(D) to describe line geometry in the Euclidean plane and P(M) to describe it for Lobachevski's plane. Yaglom's text A Simple Non-Euclidean Geometry appeared in English in 1979. There in pages 174 to 200 he develops Minkowskian geometry and describes P(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.