On Steinitz properties
type: Item
instance of: scholarly article
title: On Steinitz properties
zbMATH Open document ID: 0766.16005
DOI: 10.1007/BF01196854
published in: Archiv der Mathematik
publication date: 1992-09-27T00:00:00Z
review text: The authors are interested in discussing various `Steinitz properties' of rings \(R\) with identity, their interrelationships and connections with `other' classes of rings described in different manners. Steinitz properties of rings \(R\) with identity are variants of the following pattern (modules are unital): \(R\) is \((A)(B)\) Steinitz if any \((C)\) subset of a \((D)\) free \((A)\) \(R\)-module \(F\) can be extended to a basis by adjoining elements of an \((E)\) subset. Thus, in the original definition of left Steinitz rings one has: \((A)\equiv\) left; \((B)\) unused; \((C)\) linearly independent; \((D)\) unused; \((E)\equiv\) given basis. It was originally expected that left Steinitz rings would turn out to be division rings. Instead the class of perfect local rings turned up, demonstrating also that the intuitive idea that these rings are very suitable for linear algebra and associated geometry is quite correct. The authors of the present paper expand on this idea in a variety of ways. As one example only, if \((A)\equiv\) left, \((B)\equiv\) weakly semi, \((C)\equiv\) finite linearly independent, \((D)\equiv\) finitely generated, \((E)\equiv F\), then the resulting class is extensive as demonstrated (weakly left semi-Steinitz rings) and it has the right properties for finite dimensional linear algebra and geometry. Thus, if it is commutative Noetherian, then it is its own total quotient ring. As a demonstration of the ubiquity of these rings, it is also demonstrated that every reduced \(n\)-dimensional local ring is isomorphic to a subring of a reduced \(n\)-dimensional semi-Steinitz ring (take \((E)\equiv\) given basis instead of \((E)\equiv F)\). Altogether, the paper indicates a rich potential for further development in this entire area, the present paper being an interesting and valuable gathering of results on the way.
Mathematics Subject Classification ID: 16D80
Mathematics Subject Classification ID: 16L30
Mathematics Subject Classification ID: 16S15
Mathematics Subject Classification ID: 13C10
Mathematics Subject Classification ID: 16S10
Mathematics Subject Classification ID: 18G05
zbMATH DE Number: 62235
zbMATH Keywords: finite linearly independent
zbMATH Keywords: finitely generated
zbMATH Keywords: weakly left semi- Steinitz rings
zbMATH Keywords: reduced \(n\)-dimensional local ring
zbMATH Keywords: reduced \(n\)- dimensional semi-Steinitz ring
zbMATH Keywords: Steinitz properties
zbMATH Keywords: left Steinitz rings
zbMATH Keywords: perfect local rings
author: Budh Singh Nashier
author: Warren D. Nichols
reviewed by: J. Neggers
MaRDI profile type: Publication
cites work: Q4056148
cites work: Finitistic Dimension and a Homological Generalization of Semi-Primary Rings
cites work: On the Extension of Linearly Independent Subsets of Free Modules to Bases
cites work: Local Rings with Left Vanishing Radical
cites work: Projective modules
cites work: Serre's conjecture
cites work: A Homological Characterization of Steinitz Rings
cites work: Projective Modules Over Rings with Many Units
description: scientific article; zbMATH DE number 62235