Inversive Ring Geometry

In mathematics, inversive ring geometry is the extension of the concepts of projective line, homogeneous coordinates, projective transformations, and cross-ratio to the context of associative rings, concepts usually built upon rings that happen to be fields.

One begins with ordered pairs (a, b) in A×A where A is an (associative) ring with 1. Let U be the group of units of the ring. When there is g in U such that

(g a, g b) = (u, v),

then we write

(u, v) ~ (a, b).

In other words, we identify orbits under the action of U on the left, and ~ is the corresponding equivalence relation.

Two elements of a ring are relatively prime if the ideal in A that they generate is the whole of A. The projective line over A is the set of equivalence classes for ~ on pairs of relatively prime elements :

P(A) = { U(a, b) ∈ A × A / ~ : A a + A b = A }.