  We discuss representations of the projective line over a ring $R$ with 1 in a
projective space over some (not necessarily commutative) field $K$. Such a
representation is based upon a $(K,R)$-bimodule $U$. The points of the
projective line over $R$ are represented by certain subspaces of the projective
space $P(K,U\times U)$ 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.

Algebraic Geometry;Rings and Algebras