Lines in 3-space. (English) Zbl 1175.51002

The authors identify lines in the three-dimensional projective space \({\mathbb P}_3(F)\) over a (possibly non-commutative) field \(F\) with elements of the projective line over the ring \(M_2(F)\) of \(2 \times 2\)-matrices with entries in \(F\). Within this framework, they study reguli and regular spreads in \({\mathbb P}_3(F)\). Most result refer to the relation of regular spreads and quadratic field extensions of \(F\).
The article makes some efforts to be self-contained: the list of references is detailed, all fundamental concepts and notions are explained, and proofs for known theorems are provided.


51A45 Incidence structures embeddable into projective geometries
51A30 Desarguesian and Pappian geometries
51B05 General theory of nonlinear incidence geometry
51C05 Ring geometry (Hjelmslev, Barbilian, etc.)