The coordinatization of affine planes by rings (Q1922817)

Classical examples of affine planes consist of the points of a two-dimensional vector space, and the cosets of one-dimensional subspaces as lines. The authors define ``generalized affine planes''. Examples can be constructed via free modules of rank 2 rather than vector spaces. The authors state a Desargues type condition that characterizes generalized affine planes which can be represented by modules of rank 2. As in the classical case, to show the sufficiency of the Desargues condition is the hard part of the proof. Although there are similarities to the classical characterization of Desarguesian planes, the proof is not at all a trivial or straightforward generalization.