Rings on quotient divisible abelian groups
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911141035245568 |
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| author | E., Kompantseva T, Nguyen T. Q. |
| author_facet | E., Kompantseva T, Nguyen T. Q. |
| contents | The paper is devoted to the study of absolute ideals of groups in the class $\mathcal{QD}1$, which consists of all quotient divisible abelian groups of torsion-free rank 1. A ring is called an $AI$-ring (respectively, an $RF$-ring) if it has no ideals except absolute ideals (respectively, fully invariant subgroups) of its additive group. An abelian group is called an $RAI$-group (respectively, an $RFI$-group) if there exists at least one $AI$-ring (respectively, $FI$-ring) on it. If every absolute ideal of an abelian group is a fully invariant subgroup, then this group is called an $afi$-group.
It is shown that every group in $\mathcal{QD}1$ is an $RAI$-group, an $RFI$-group, and an $afi$-group. Thus, Problem 93 of L. Fuchs' monograph \emph{``Infinite Abelian Groups, Vol. II, New York-London: Academic Press, 1973''} is resolved within the class $\mathcal{QD}1$. For any group in $\mathcal{QD}1$, all rings on it that are $AI$-rings are described. Furthermore, the set of all $AI$-rings on $G \in \mathcal{QD}1$ coincides with the set of all $FI$-rings on $G$. In addition, the principal absolute ideals of groups in $\mathcal{QD}1$ are described. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_05572 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Rings on quotient divisible abelian groups E., Kompantseva T, Nguyen T. Q. Group Theory Rings and Algebras 20K30, 20K99, 16B99 The paper is devoted to the study of absolute ideals of groups in the class $\mathcal{QD}1$, which consists of all quotient divisible abelian groups of torsion-free rank 1. A ring is called an $AI$-ring (respectively, an $RF$-ring) if it has no ideals except absolute ideals (respectively, fully invariant subgroups) of its additive group. An abelian group is called an $RAI$-group (respectively, an $RFI$-group) if there exists at least one $AI$-ring (respectively, $FI$-ring) on it. If every absolute ideal of an abelian group is a fully invariant subgroup, then this group is called an $afi$-group. It is shown that every group in $\mathcal{QD}1$ is an $RAI$-group, an $RFI$-group, and an $afi$-group. Thus, Problem 93 of L. Fuchs' monograph \emph{``Infinite Abelian Groups, Vol. II, New York-London: Academic Press, 1973''} is resolved within the class $\mathcal{QD}1$. For any group in $\mathcal{QD}1$, all rings on it that are $AI$-rings are described. Furthermore, the set of all $AI$-rings on $G \in \mathcal{QD}1$ coincides with the set of all $FI$-rings on $G$. In addition, the principal absolute ideals of groups in $\mathcal{QD}1$ are described. |
| title | Rings on quotient divisible abelian groups |
| topic | Group Theory Rings and Algebras 20K30, 20K99, 16B99 |
| url | https://arxiv.org/abs/2509.05572 |