and b in Z such that ap + bq = 1, so that U[p, q] is in P(Z/nZ) but it is not an image of an element under the canonical embedding. The whole of P(Z/nZ) is filled out by elements U[up, vq], u ≠ v, u, v ∈ U = the units of Z/nZ. The instances Z/nZ are given here for n = 6, 10, and 12, where according to modular arithmetic the group of units of the ring is U = {1, 5}, U = {1,3,7,9}, and U = {1,5,7,11} respectively. Modular arithmetic will confirm that, in each table, a given letter represents multiple points. In these tables a point U[m, n] is labeled by m in the row at the table bottom and n in the column at the left of the table. For instance, the point at infinity A = U[v, 0], where v is a unit of the ring. The extra points can be associated with Q ⊂ R ⊂ C, the rationals in the extended complex upper-half plane. The group of homographies on P(Z/nZ) is called a principal congruence subgroup. For the rational numbers Q, homogeneity of coordinates means that every element of P(Q) may be represented by an element of P(Z). Similarly, a homography of P(Q) corresponds to an element of the modular group, the automorphisms of P(Z). Over continuous rings The projective line over a division ring results in a single auxiliary point . Examples include the real projective line, the complex projective line, and the projective line over quaternions. These examples of topological rings have the projective line as their one-point compactifications. The case of the complex number field C has the Möbius group as its homography group. The projective line over the dual numbers was described by Josef Grünwald in 1906. This ring includes a nonzero nilpotent n satisfying . The plane of dual numbers has a projective line including a line of points . Isaak Yaglom has described it as an "inversive Galilean plane" that has the topology of a cylinder when the supplementary line is included. Similarly, if A is a local ring, then P(A) is formed by adjoining points corresponding to the elements of the maximal ideal of A. The projective line over the ring M of split-complex numbers introduces auxiliary lines and Using stereographic projection the plane of split-complex numbers is closed up with these lines to