Affine geometry of modules over a ring with an invariant basis number
The validity of the fundamental theorem for affine geometries over rings with invariant basis number is investigated. For a free \(R\)--module \(_RM\), the affine space can be interpreted in different ways: as the set of all unimodular elements, as the set of all \(e\) for which \(Re\) is a free, one-dimensional submodule and as a complete semimodular lattice of cosets. According to the different interpretations, some versions of the fundamental theorem are proved. It turns out that isomorphisms between affine geometries over the rings \(R\) and \(R_1\) correspond to semilinear mappings between the underlying modules over those rings.
scientific article; zbMATH DE number 5637525
scientific article