A characterization of \(\aleph _ 1\)-free Abelian groups and its application to the Chase radical
type: Item
instance of: scholarly article
title: A characterization of \(\aleph _ 1\)-free Abelian groups and its application to the Chase radical
zbMATH Open document ID: 0655.20038
DOI: 10.1007/BF02766167
author: Katsuya Eda
published in: Israel Journal of Mathematics
publication date: 1987-01-01T00:00:00Z
review text: For any complete Boolean algebra B and an (abelian) group A the Boolean power \(A^{(B)}\) is the group of functions f: \(A\to B\) such that \(\{\) f(a); \(a\in A\}\) is a partition of 1 in B and the sum \(f+g\) maps \(a\in A\) onto \(\bigvee_{x\in A}f(x)\wedge g(a-x)\). The main results of the paper are: 1. A group A is \(\aleph_ 1\)-free (i.e. all its countable subgroups are free) iff A is a subgroup of the Boolean power \(Z^{(B)}\) for some B. 2. The Chase radical \(\nu (A)=\cap \{\ker h\); \(h\in Hom(A,X)\), X is \(\aleph_ 1\)-free\(\}\) is equal to \(\sum \{C\subseteq A\); \(Hom(C,Z)=0\), C is countable\(\}\). 3. The torsion class which consists of all groups A such that \(\nu (A)=A\) is not closed under uncountable direct products (it is known to be closed under countable direct products).
Mathematics Subject Classification ID: 20K25
Mathematics Subject Classification ID: 20K20
Mathematics Subject Classification ID: 20K27
zbMATH DE Number: 4069196
zbMATH Keywords: \(\aleph _ 1\)-free groups
zbMATH Keywords: complete Boolean algebra
zbMATH Keywords: Boolean power
zbMATH Keywords: Chase radical
zbMATH Keywords: torsion class
zbMATH Keywords: direct products
reviewed by: Stanisław Balcerzyk
MaRDI profile type: Publication
cites work: On groups of functions defined on Boolean algebras
cites work: Q4146730
cites work: On reduced products of Abelian groups
cites work: On radicals and products
cites work: A Boolean power and a direct product of Abelian groups
cites work: Q3667083
cites work: On a Boolean power of a torsion free Abelian group
cites work: Q4205649
cites work: Compact cardinals and Abelian groups
cites work: On Boolean powers of the group Z and (\(\omega\) ,\(\omega\) )-weak distributivity
cites work: Q3789783
cites work: Categoricity regained
cites work: Q3667082
cites work: Summands of Separable Abelian Groups: Dedicated to Paul Turán on his 60th Birthday
cites work: Note on extensions of Abelian groups by primary groups
cites work: Q3852172
cites work: On the slender property of certain Boolean algebras
cites work: An extension of a theorem of Gaifman-Hales-Solvay
cites work: Countable approximations and Löwenheim-Skolem theorems
cites work: Set theory. An introduction to independence proofs
cites work: Q5585437
cites work: Q3324156
description: scientific article; zbMATH DE number 4069196