Abstract
This chapter presents the theory of projective planes and spaces over associative rings of stable rank 2, from both an algebraic and an axiomatic point of view. It deals with homomorphisms of these and treats the projective spaces over full matrix rings as an example; they can be interpreted in terms of ordinary projective spaces over division rings. The chapter discusses spaces over local rings (Klingenberg spaces) and over Hjelmslev rings (Hjelmslev spaces), presented as special instances of the general theory together with a few other special cases. It describes transvection planes, Faulkner planes, affine planes over two-sided units rings, and their axiomatic characterization as Desarguesian Leißner planes, and a short list of open problems is given. For an axiomatic characterization of projective spaces over rings, the notion of Barbilian space is introduced; this is a set of points and a set of hyperplanes together with an incidence and a neighbor relation.
Published Version
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