Strongly and Uniformly Strongly co-Hopfian Abelian Groups
Andrey R. Chekhlov, Peter V. Danchev, Patrick W. Keef
TL;DR
The paper investigates strongly co-Hopfian abelian groups, introducing a splitting criterion that reduces analyses to reduced parts. It proves that if the torsion subgroup T and the quotient G/T are Sco-H, then G is Sco-H, while the converse can fail, highlighting the complexity of mixed groups. It further connects Sco-H to cotorsion and algebraic compactness, shows that uniform Sco-H corresponds to the maximal torsion subgroup being Sco-H, and provides explicit decomposition criteria for mixed groups, together with counterexamples that delineate the limits of these results. Overall, the work advances the structure theory of Sco-H groups by clarifying when torsion and torsion-free components dictate global co-Hopficity and by outlining open problems about direct sums of Sco-H groups.
Abstract
We consider the so-called {\it strongly co-Hopfian} and {\it uniformly strongly co-Hopfian} Abelian groups, significantly generalizing some important results due to Abdelalim in the J. Math. Analysis (2015). Specifically, we prove that any strongly co-Hopfian group is a direct sum of an sp-group and a divisible group, both of which are strongly co-Hopfian. We also show that a group whose maximal torsion subgroup and corresponding torsion-free factor are both strongly co-Hopfian will also be strongly co-Hopfian. We provide several examples demonstrating that the converse of this statement does {\it not} generally hold, thus illustrating that the structure of genuinely mixed strongly co-Hopfian groups is rather complicated and does {\it not} entirely depend on the structure of its maximal torsion subgroup. We also establish that a strongly co-Hopfian group is cotorsion exactly when it is algebraically compact and, particularly, a reduced (adjusted) cotorsion group is strongly co-Hopfian only when its maximal torsion subgroup is strongly co-Hopfian. Additionally, we demonstrate that a strongly co-Hopfian group is uniformly strongly co-Hopfian exactly when its maximal torsion subgroup is strongly co-Hopfian.