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 homogeneous 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 .
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