Abstract
We introduce the chain geometry Σ(K,R) over a ring R with a distinguished subfield K, thus extending the usual concept where R has to be an algebra over K. A chain is uniquely determined by three of its points, if, and only if, the multiplicative group of K is normal in the group of units of R. This condition is not equivalent to R being a K-algebra. The chains through a fixed point fall into compatibility classes which allow to describe the residue at a point in terms of a family of affine spaces with a common set of points.
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Blunck, A., Havlicek, H. Extending the Concept of Chain Geometry. Geometriae Dedicata 83, 119–130 (2000). https://doi.org/10.1023/A:1005260729790
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DOI: https://doi.org/10.1023/A:1005260729790