Semi-rigid classes of cotorsion-free Abelian groups
type: Item
instance of: scholarly article
title: Semi-rigid classes of cotorsion-free Abelian groups
zbMATH Open document ID: 0554.20018
DOI: 10.1016/0021-8693(85)90178-4
author: Rüdiger Göbel
author: S. Shelah
published in: Journal of Algebra
publication date: 1985-01-01T00:00:00Z
review text: A class of groups is semi-rigid if for every pair (H,G) in it, either Hom(H,G) or \(Hom(G,H)=0\), and rigid if both are zero. The authors use combinatorics of infinite sets to prove powerful theorems on the existence of large semi-rigid families (in ZFC) and large rigid families (in ZFC \(+\) additional axioms). The main results depend on the following theorem: for every strong limit singular cardinal m of uncountable cofinality, there is a cotorsion-free group G of cardinal m such that every proper subgroup U with cotorsion- free quotient has smaller cardinality. Moreover G may be chosen to be homogeneous of any non-divisible type. As applications of these results, the authors prove several results about torsion theories; for example the class of cotorsion-free groups does not have a cogenerating set, and torsion classes generated or cogenerated by a single group do not have small cardinality.
reviewed by: Phillip Schultz
Mathematics Subject Classification ID: 20K20
Mathematics Subject Classification ID: 20K27
Mathematics Subject Classification ID: 03E55
zbMATH DE Number: 3882638
zbMATH Keywords: large semi-rigid families
zbMATH Keywords: ZFC
zbMATH Keywords: large rigid families
zbMATH Keywords: strong limit singular cardinal
zbMATH Keywords: cotorsion-free group
zbMATH Keywords: torsion classes
MaRDI profile type: Publication
full work available at URL: 0021-8693(85)90178-4
OpenAlex ID: W2041043898
cites work: Endomorphism algebras of large modules with distinguished submodules
cites work: On torsion classes of Abelian groups
cites work: A Torsion Theory for Abelian Categories
cites work: Every Cotorsion-Free Ring is an Endomorphism Ring
cites work: On endomorphism rings of primary Abelian groups
cites work: Every cotorsion-free algebra is an endomorphism algebra
cites work: Arbitrary torsion classes and almost free abelian groups
cites work: Arbitrary torsion classes of abelian groups*
cites work: On constructing indecomposable groups in L
cites work: Intersection Theorems for Systems of Sets (II)
cites work: Q4093653
cites work: Q5673086
cites work: On stout and slender groups
cites work: Solution of two problems on cotorsion abelian groups
cites work: Q3861604
cites work: Lösung eines Problems von L. Fuchs
cites work: Some aspects of torsion
cites work: Q3852172
cites work: The fine structure of the constructible hierarchy
cites work: Q4159049
cites work: Torsion theories, additive semantics, and rings of quotients. With an appendix by H. H. Storrer on torsion theories and dominant dimensions
cites work: How to Construct Almost Free Groups
cites work: Q4177758
cites work: Infinite Abelian groups, Whitehead problem and some constructions
cites work: Q4095071
cites work: Q3879436
cites work: On endo-rigid, strongly \(\aleph_1\)-free Abelian groups in \(\aleph_1\).
cites work: Q3704092
cites work: Diamonds, uniformization
cites work: Coherent Rings and Fp -Injective Modules
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description: scientific article; zbMATH DE number 3882638