Extending the Concept of Chain Geometry

We introduce the chain geometry (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.