On classifying affine Barbilian spaces
type: Item
instance of: scholarly article
title: On classifying affine Barbilian spaces
zbMATH Open document ID: 0634.51005
DOI: 10.1007/BF03322386
published in: Results in Mathematics
publication date: 1987-01-01T00:00:00Z
review text: The author considers the representation of an affine Barbilian space by means of the corresponding affine geometry over a unitary free module, i.e. \(A_{FF}(M_ R,B)=(P,L,\phi,\|)\), where \(M_ R\) is a free unitary right R-module over an arbitrary ring R with identity, B is a Barbilian domain; \(P=set\) of points, \(L=set\) of lines, \(\phi =relation\) of non-neighbouredness, \(\| =parallelism\). In this way, he characterizes algebraic properties by geometric ones, and viceversa. The following are two of the results: a) \((P,L,\phi,\|)\) is linear unique (any two different points are contained in at most one line) iff R has no zero-divisors. - b) \((P,L,\phi,\|)\) is linear connected (any two different points are contained in at least one line) iff \(BR=M_ R.\) He also proves that the subclass of affine Hjelmslev spaces is characterized by the transivity of the neighboured relation, i.e. complementary to \(\phi\).
Mathematics Subject Classification ID: 51C05
zbMATH DE Number: 4032489
zbMATH Keywords: parallel
zbMATH Keywords: non-neighboured
zbMATH Keywords: weakly 1-finite ring
zbMATH Keywords: affine Barbilian space
zbMATH Keywords: linear unique
zbMATH Keywords: zero-divisors
zbMATH Keywords: linear connected
author: Werner Leissner
reviewed by: Salvador D. Gigena
MaRDI profile type: Publication
cites work: Ebene Geometrie über einem Ring
cites work: On Barbilian domains over commutative rings
cites work: On the structure of the \(GL_ 2\) of a ring
cites work: Q5639812
cites work: Affine geometry over free unitary modules
cites work: Affine Barbilian structures
full work available at URL: https://doi.org/10.1007/bf03322386
OpenAlex ID: W2076213259
description: scientific article; zbMATH DE number 4032489