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
over a commutative ring
with unit 1 and
, 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
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)].