Abstract
This article surveys the theory of projective planes over rings of stable rank 2. Such a plane is described as a structure of points and lines together with an incidence relation and a neighbor relation and which has to satisfy two groups of axioms. The axioms in the first group express elementary relations between points and lines such as, e.g., the existence of a unique line joining any two non-neighboring points, and define what is called a Barbilian plane. In the second group of axioms the existence of sufficiently many transvections, dilatations, and generalizations of the latter, the affine dilatations and their duals, is required. Additional geometric properties of planes over special types of rings are then discussed. The paper ends with the treatment of homomorphisms between ring planes, i.e., of (not necessarily bijective) mappings which preserve incidence.
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Veldkamp, F.D. (1985). Projective Ring Planes and Their Homomorphisms. In: Kaya, R., Plaumann, P., Strambach, K. (eds) Rings and Geometry. NATO ASI Series, vol 160. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-5460-1_6
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DOI: https://doi.org/10.1007/978-94-009-5460-1_6
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