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) = { U[a, b] : aA + bA = A }, that is, U[a, b] is in the projective line if the ideal generated by a and b is all of A. The projective line P(A) is equipped with a group of homographies. The homographies are expressed through use of the matrix ring over A and its group of units V as follows: If c is in Z(U), the center of U, then the group action of matrix on P(A) is the same as the action of the identity matrix. Such matrices represent a normal subgroup N of V. The homographies of P(A) correspond to elements of the quotient group V / N . P(A) is considered an extension of the ring A since it contains a copy of A due to the embedding E : a → U[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,v ∈ U, the mapping a → uav 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. 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. 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.