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 isomorphic to their complements. Cross-ratio 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 the cross-ratio is taken as the value of h on a fourth point . To build h from a, b, c the generator homographies 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 , , are all units, then is a unit. proof: Evidently is a unit, as required. Theorem: If is a unit, then there is a homography h in G(A) such that h(a) = U[0,1], h(b) = U[1,1], and h(c) = U[1,0]. proof: The point is the image of b after a was put to 0 and then inverted to U[1,0], and the image of c is brought to U[0,1]. As p is a unit, its inverse used in a rotation will move p to U[1,1], resulting in a, b, c being all properly placed. The lemma refers to sufficient conditions for the existence of h. 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. History 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