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The geometry of null systems, Jordan algebras and von Staudt’s theorem. (English) Zbl 1038.17023
The paper under review extends definitions and results on generalized projective and polar geometry and their equivalence with the Jordan structures of the author’s recent paper [Adv. Geom. 2, 329–369 (2002; Zbl 1035.17043)]. The main topic of this work is that the generalization of a connected generalized projective geometry $\left(X,{X}^{\text{'}}\right)$ over a commutative ring $K$ with unit 1 and $\frac{1}{2}\in K$, is given by spaces corresponding to unital Jordan algebras, that is Jordan pair together with a distinguished invertible element. Also, the geometric interpretation of unital Jordan algebras is discussed. Precisely, there is canonically associated to the geometry $\left(X,{X}^{\text{'}}\right)$ a class of symmetric spaces. Additionally, a generalization is given to the well-known von Staudt’s theorem [M. Berger, Geometry I, II. Berlin: Springer-Verlag (1987; Zbl 0606.51001), Reprint Springer (1994)].
##### MSC:
 17C37 Associated geometries 51A05 General theory of linear incidence geometry; projective geometries 53C35 Symmetric spaces (differential geometry)