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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.)
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