Determination of a class of countable-rank, torsion-free Abelian groups by their endomorphism rings.
It has been previously shown that if \(X\) and \(Y\) are block-rigid, almost completely decomposable torsion-free Abelian groups of ring type and finite rank with cyclic regulator quotient, then \(X\) and \(Y\) are near isomorphic if and only if \(\text{End}(X)\) and \(\text{End}(Y)\) are isomorphic. The current work extends this result to block-rigid, `local' almost completely decomposable torsion-free Abelian groups of ring type and `countable' rank with cyclic regulator quotient. A block-rigid almost completely decomposable group with cyclic regulator quotient is a torsion-free Abelian group \(X\) containing a uniquely determined, completely decomposable, fully invariant subgroup \(A=R(X)\) called its regulator such that \(X/A\) is a finite cyclic group. The definition of a local almost completely decomposable group with cyclic regulator quotient is more complicated, but is included in the paper under review.
almost completely decomposable block rigid torsion-free Abelian groups
scientific article; zbMATH DE number 5358195