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 P¹(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].

P¹(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 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(U), the center of U, then the group action of matrix ${\displaystyle \left({\begin{smallmatrix}c&0\\0&c\end{smallmatrix}}\right)}$ 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 the group of units U of A, is expressed by a homography on P¹(A):

${\displaystyle U[a,1]{\begin{pmatrix}0&1\\1&0\end{pmatrix}}=U[1,a]\thicksim U[a^{-1},1].}$

Furthermore, for u,v ∈ U, the mapping a → uav can be extended to a homography:

${\displaystyle {\begin{pmatrix}u&0\\0&1\end{pmatrix}}{\begin{pmatrix}0&1\\1&0\end{pmatrix}}{\begin{pmatrix}v&0\\0&1\end{pmatrix}}{\begin{pmatrix}0&1\\1&0\end{pmatrix}}={\begin{pmatrix}u&0\\0&v\end{pmatrix}}.}$

${\displaystyle U[a,1]{\begin{pmatrix}v&0\\0&u\end{pmatrix}}=U[av,u]\thicksim U[u^{-1}av,1].}$

Since u is arbitrary, it may be substituted for u⁻¹. Homographies on P¹(A) are called linear-fractional transformations since

${\displaystyle U[z,1]{\begin{pmatrix}a&c\\b&d\end{pmatrix}}=U[za+b,zc+d]\thicksim U[(zc+d)^{-1}(za+b),1].}$