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<p>We consider the so-called <i>strongly co-Hopfian</i> and <i>uniformly strongly co-Hopfian</i> 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 <i>not</i> generally hold, thus illustrating that the structure of genuinely mixed strongly co-Hopfian groups is rather complicated and does <i>not</i> 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.</p>

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Strongly and uniformly strongly co-Hopfian Abelian groups

  • 期刊论文

  • 发表日期：

    2026年5月22日

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  • Andrey R. Chekhlov,
  • Peter V. Danchev,
  • Patrick W. Keef

摘要

We consider the so-called strongly co-Hopfian and 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 not generally hold, thus illustrating that the structure of genuinely mixed strongly co-Hopfian groups is rather complicated and does 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.