Reguli and chains over skew fields (Q1975381)

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 \textit{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 \(U\oplus U\cong V\).