Two problems on \(\aleph _ 0\)-indecomposable Abelian groups
An abelian group G is \(\aleph_ 0\)-indecomposable if G cannot be written as a direct sum of \(\aleph_ 0\) non-zero groups. A convenient but interesting reformulation of the \(\aleph_ 0\)-decomposability for a reduced countable mixed group G with finite torsion-free rank is that every finite subset can be imbedded in a summand of finite index. Using this the authors show that a countable reduced torsion group T is the torsion part of an \(\aleph_ 0\)-indecomposable mixed group G with torsion-free rank n exactly when \(T=F\oplus H\), where F is finite, and for each prime p and positive integer k, \(f^ b_ k(A)+f^ b_{k+1}(A)\leq n\) where \(f^ b_ k(A)\) is the kth Ulm-Kaplansky invariant of the p-component \(H_ p\).
\(\aleph _ 0\)-indecomposable mixed group
scientific article; zbMATH DE number 3961974