Projective planes over rings
type: Item
instance of: scholarly article
title: Projective planes over rings
zbMATH Open document ID: 0632.51007
DOI: 10.1007/BF03322399
author: Frieder Knüppel
published in: Results in Mathematics
publication date: 1987-01-01T00:00:00Z
review text: The author considers associative rings R with a unit element and defines a generalized projective plane p(R), based on four axioms adapted from \textit{F. D. Veldkamp} [Geom. Dedicata 11, 285-308 (1981; Zbl 0468.51001)] except one that is replaced by a weaker property. A coordinatization theorem is stated but without proof due to its length. This theorem is not quite satisfying since two additional assumptions are required, which are not fulfilled in every projective ring plane p(R). One is valid in such a plane if R is commutative and the stable rank of R is less than or equal to three. An example of a projective ring plane is given where this property does not hold.
Mathematics Subject Classification ID: 51C05
zbMATH DE Number: 4028265
zbMATH Keywords: associative rings with a unit element
zbMATH Keywords: axioms
zbMATH Keywords: coordinatization
zbMATH Keywords: projective ring plane
zbMATH Keywords: stable rank
reviewed by: S. R. Mandan
MaRDI profile type: Publication
cites work: Q3241211
cites work: Q3254794
cites work: Ebene Geometrie über einem Ring
cites work: Desarguessche Ebenen mit Nachbarelementen
cites work: Äquiforme Ebenen über kommutativen Ringen und singuläre Prä-Hjelmslev-Gruppen
cites work: Plane elliptic geometry over rings
cites work: Singuläre Hjelmslev-Gruppen
cites work: Einbettung von Hjelmslev-Gruppen in orthogonale Gruppen über kommutativen Ringen
cites work: Stable rank of rings and dimensionality of topological spaces
cites work: Projective planes over rings of stable rank 2
description: scientific article; zbMATH DE number 4028265