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 × 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. Mathematics Subject Classification (1991): 51C05, 51A45, 51B05.

The set G of all m-dimensional subspaces of a 2m-dimensional vector space V is endowed with two relations, complementarity and adjacency. We consider bijections from G onto G ′ , where G ′ arises from a 2m ′ -dimensional vector space V ′ . If such a bijection ϕ and its inverse leave one of the relations from above invariant, then also the other. In case m ≥ 2 this yields that ϕ is induced by a semilinear bijection from V or from the dual space of V onto V ′ .As far as possible, we include also the infinite-dimensional case into our considerations.2000 Mathematics Subject Classification: 51A10, 51A45, 05C60.

We define generalized Clifford parallelisms in PG(3, F) with the help of a quaternion skew field H over a field F of arbitrary characteristic. Moreover we give a geometric description of such parallelisms involving hyperbolic quadrics in projective spaces over suitable quadratic extensions of F .

The main result of the present paper is that the projective line over a ring R is connected with respect to the relation "distant" if, and only if, R is a GE 2 -ring. Mathematics Subject Classification (2000): 51C05, 20H25.

We determine all distant-isomorphisms between projective lines over semilocal rings. In particular, for those semisimple rings that do not have a simple component which is isomorphic to a field, every distant isomorphism arises from a Jordan isomorphism of rings and a projectivity. We show this by virtue of a one-one correspondence linking the projective line over a semisimple ring with a Segre product of Grassmann spaces. (2000). 51C05, 51A10, 51A45, 17C50. Mathematics Subject Classification