Countable ℵ0-indecomposable mixed abelian groups of finite torsion-free rank
Copyright policy )Countable ℵ0-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 \(B_ i\).
- University of Colorado System United States
- University of Colorado Boulder United States
- Hebrew University of Jerusalem Israel
- Department of Mathematics
- The Hebrew University Israel
Richman type, Algebra and Number Theory, extension, direct sum, reduced torsion abelian group, basic subgroup, Direct sums, direct products, etc. for abelian groups, Mixed groups, \(\aleph _ 0\)-indecomposable, finite rank torsion-free abelian group, Ulm-Kaplansky invariants, Abelian groups
Richman type, Algebra and Number Theory, extension, direct sum, reduced torsion abelian group, basic subgroup, Direct sums, direct products, etc. for abelian groups, Mixed groups, \(\aleph _ 0\)-indecomposable, finite rank torsion-free abelian group, Ulm-Kaplansky invariants, Abelian groups
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