Strongly and Uniformly Strongly co-Hopfian Abelian Groups
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arXiv
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| Natura: | Preprint |
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2025
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| author | Chekhlov, Andrey R. Danchev, Peter V. Keef, Patrick W. |
| author_facet | Chekhlov, Andrey R. Danchev, Peter V. Keef, Patrick W. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_22268 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Strongly and Uniformly Strongly co-Hopfian Abelian Groups Chekhlov, Andrey R. Danchev, Peter V. Keef, Patrick W. Group Theory Rings and Algebras 20K10, 20K20, 20K21, 20K30 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. |
| title | Strongly and Uniformly Strongly co-Hopfian Abelian Groups |
| topic | Group Theory Rings and Algebras 20K10, 20K20, 20K21, 20K30 |
| url | https://arxiv.org/abs/2511.22268 |