Projective representations i. projective lines over rings

We discuss representations of the projective line over a ringR with 1 in a projective space over some (not necessarily commutative) fieldK. Such a representation is based upon a (K, R)-bimoduleU. The points of the projective line overR are represented by certain subspaces of the projective space ℙ(K, U ×U) that are isomorphic to one of their complements. In particular, distant points go over to complementary subspaces, but in certain cases, also non-distant points may have complementary images.