stylistic:style_transfer, stylistic:tone
Over a ring

Suppose A is a ring and U is its group of units. Homographies act on a projective line over A, written P(A), consisting of points U with projective coordinates. The homographies on P(A) are described by matrix mappings

When A is a commutative ring, the homography may be written

but otherwise the linear fractional transformation is seen as an equivalence:

The homography group of the ring of integers Z is modular group PSL(2, Z). Ring homographies have been used in quaternion analysis, and with dual quaternions to facilitate screw theory. The conformal group of spacetime can be represented with homographies where A is the composition algebra of biquaternions.