## 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.

### MSC:

 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.)

### Keywords:

projective line over a ring; regulus; spread; Baer subspace