Projective line over a ring

Summary

The projective line over a ring $R$ extends the usual projective line over a field to any associative ring with identity. Its points are the pairs $(a, b) \in R^2$ that can be completed to an invertible $2 \times 2$ matrix, taken up to multiplication on the left by units. The group $\mathrm{GL}(2, R)$ acts on it. The construction gives models of circle geometries, such as the Möbius plane from the complex numbers and the Minkowski plane from the split-complex numbers.