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Reguli and chains over skew fields. (English) Zbl 0952.51002

The author gives a new geometric definition of a regulus in a not necessarily pappian projective space \(PG(V,K)\) of arbitrary dimension. She proves that any three mutually complementary subspaces are contained in a regulus, which is unique precisely if the skew field \(K\) is commutative. Furthermore, these reguli have an interpretation in chain geometry [compare the author, Bull. Belg. Math. Soc. Simon Stevin 6, No. 4, 589-603 (1999)] and W. Benz [‘Vorlesungen über Geometrie der Algebren’, Springer Verlag, Berlin (1973; Zbl 0258.50024)]: there is a natural bijection which maps reguli onto the chains of the projective line over the ring \(R=\text{End}_K (U)\), where \(U\oplus U\cong V\).

MSC:

51A30 Desarguesian and Pappian geometries
51B05 General theory of nonlinear incidence geometry
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