Countable \(\aleph _ 0\)-indecomposable mixed Abelian groups of finite torsion-free rank
A group is \(\aleph_ 0\)-indecomposable if it cannot be decomposed into a direct sum of \(\aleph_ 0\) non-zero summands. In this paper the authors find necessary and sufficient conditions for an extension of a countable reduced torsion abelian group T by a finite rank torsion-free abelian group R to be \(\aleph_ 0\)-indecomposable. These conditions are remarkably simple, and depend only on T and R: Let R have rank n and Richman type the matrix t. Then T has a basic subgroup which decomposes as a direct sum of \(n+1\) summands \(B_ 0,B_ 1,...,B_ n\) such that \(B_ 0\) is finite; for \(i\geq 1\), \(B_ i\) has no two adjacent non-zero Ulm-Kaplansky invariants; and for \(i\geq 1\) and all primes p, \([r_{pi}]\leq t\), where \(r_{pi}\) is the p-rank of \(B_ i\).
scientific article; zbMATH DE number 3939591
scientific article