A characterization of \(\aleph _ 1\)-free Abelian groups and its application to the Chase radical
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).
scientific article; zbMATH DE number 4069196
scientific article