Let R be a ring with 1, and let K be a not necessarily commutative subfield of R. The chain geometry S (K, R) is
an incidence structure consisting of points and chains: The point set is the projective line over R, and the chains are
the images of the projective line over K under the group GL 2 (R).
Up to now, chain geometries have mainly been studied for the case that K lies in the centre of R (and thus is
commutative). Then S (K, R) is a so-called chain space. In this project we aim at an appropriate generalization of
the synthetic concept of a chain space. So we want to establish an axiom system for a class of incidence structures
containing all chain geometries over skew fields and all chain spaces.
Explicit examples like the generalized chain spaces arising from reguli in not necessarily pappian projective spaces
shall be investigated. Moreover, we want to introduce the notions of residual spaces and of subspaces of
generalized chain spaces. They shall be studied for both the analytic examples S (K, R) and the geometric examples
mentioned above.