Affine geometry of modules over a ring with an invariant basis number
type: Item
instance of: scholarly article
title: Affine geometry of modules over a ring with an invariant basis number
zbMATH Open document ID: 1200.51003
published in: Journal of Mathematical Sciences (New York)
publication date: 2009-11-24T00:00:00Z
review text: The validity of the fundamental theorem for affine geometries over rings with invariant basis number is investigated. For a free \(R\)--module \(_RM\), the affine space can be interpreted in different ways: as the set of all unimodular elements, as the set of all \(e\) for which \(Re\) is a free, one-dimensional submodule and as a complete semimodular lattice of cosets. According to the different interpretations, some versions of the fundamental theorem are proved. It turns out that isomorphisms between affine geometries over the rings \(R\) and \(R_1\) correspond to semilinear mappings between the underlying modules over those rings.
Mathematics Subject Classification ID: 51C05
Mathematics Subject Classification ID: 51A10
Mathematics Subject Classification ID: 13C10
zbMATH DE Number: 5637525
zbMATH Keywords: affine geometry
zbMATH Keywords: invariant basis number
zbMATH Keywords: module
zbMATH Keywords: geometry over ring
author: Alexander Lashkhi
author: T. G. Kvirikashvili
reviewed by: Dirk Keppens
MaRDI profile type: Publication
full work available at URL: s10958-006-0292-6
OpenAlex ID: W2082758084
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DOI: 10.1007/S10958-006-0292-6
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description: scientific article; zbMATH DE number 5637525