a hyperboloid of one sheet. The projective line over M may be called the Minkowski plane when characterized by behaviour of hyperbolas under homographic mapping. Chains The real line in the complex plane gets permuted with circles and other real lines under Möbius transformations, which actually permute the canonical embedding of the real projective line in the complex projective line. Suppose A is an algebra over a field F, generalizing the case where F is the real number field and A is the field of complex numbers. The canonical embedding of P(F) into P(A) is A chain is the image of P(F) under a homography on P(A). Four points lie on a chain if and only if their cross-ratio is in F. Karl von Staudt exploited this property in his theory of "real strokes" [reeler Zug]. Point-parallelism Two points of P(A) are parallel if there is no chain connecting them. The convention has been adopted that points are parallel to themselves. This relation is invariant under the action of a homography on the projective line. Given three pair-wise non-parallel points, there is a unique chain that connects the three. Modules The projective line P(A) over a ring A can also be identified as the space of projective modules in the module . An element of P(A) is then a direct summand of . This more abstract approach follows the view of projective geometry as the geometry of subspaces of a vector space, sometimes associated with the lattice theory of Garrett Birkhoff or the book Linear Algebra and Projective Geometry by Reinhold Baer. In the case of the ring of rational integers Z, the module summand definition of P(Z) narrows attention to the U[m, n], m coprime to n, and sheds the embeddings which are a principal feature of P(A) when A is topological. The 1981 article by W. Benz, Hans-Joachim Samaga, & Helmut Scheaffer mentions the direct summand definition. In an article "Projective representations: projective lines over rings" the group of units of a matrix ring M2(R) and the concepts of module and bimodule are used to define a projective line over a ring. The group of units is denoted by GL(2,R), adopting notation from the general linear group, where R is usually taken to be a field. The projective line is the set of orbits under GL(2,R) of the free cyclic submodule R(1,0) of . Extending the commutative theory