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 P1(A) over A consists of points identified by projective coordinates. Let A× be the group of units of A; pairs (a, b) and (c, d) from A × A are related when there is a u in A× such that ua = c and ub = d. This relation is an equivalence relation. A typical equivalence class is written U[a, b].
P1(A) = { U[a, b] | aA + bA = A }, that is, U[a, b] is in the projective line if the one-sided ideal generated by a and b is all of A.
The projective line P1(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(A×), the center of A×, then the group action of matrix on P1(A) is the same as the action of the identity matrix. Such matrices represent a normal subgroup N of V. The homographies of P1(A) correspond to elements of the quotient group V / N.
P1(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 A×, is expressed by a homography on P1(A):
Furthermore, for u,v ∈ A×, the mapping a → uav can be extended to a homography:
Since u is arbitrary, it may be substituted for u−1. Homographies on P1(A) are called linear-fractional transformations since