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| Main Authors: | , , |
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| Format: | Preprint |
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2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2401.07742 |
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| _version_ | 1866909073529634816 |
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| author | Amini, A. Amini, B. Momtahan, E. |
| author_facet | Amini, A. Amini, B. Momtahan, E. |
| contents | Let $G$ be a sp-group such that for every prime $p$, $G_p$ is elementary. %$\oplus \End_{\zz}(G_p) \leq \End_{\zz}(G) \leq \prod \End_{\zz}(G_p)$. Suppose that $\frac{G}{\oplus_{p\in \mathbb{P}} G_p}$ is torsion-free divisible.
%In this article we characterize pure subrings of $\prod_{p\in \mathbb{P}} \End(G_p)$.
We show that $\End_{\zz}(G)$ is a sp-group and every subring $R$ of $\prod \End_{\zz}(G_p)$, containing $\oplus \End_{\zz}(G_p)$ is pure if and only if $R=\mathbb{M}_T=\{x\in \prod_{p\in \mathbb{P}}\End(G_p) \;|\; \exists k\in \nn \;\mbox{\rm{such that}} \;\; kx \in T \},$ where $T$ is a subring of $\prod_{p\in \mathbb{P}}\End(G_p)$. We observe that $\frac{\mathbb{M}_T}{\oplus_{p\in \mathbb{P}}\End(G_p)}$ is (ring) isomorphic with $T\otimes_{\zz} \qq$. Moreover, we conclude that a significant number of the examples around the topic can be easily obtained and described by choosing an appropriate subring $T$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_07742 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On pure subrings of sp-groups Amini, A. Amini, B. Momtahan, E. Group Theory Let $G$ be a sp-group such that for every prime $p$, $G_p$ is elementary. %$\oplus \End_{\zz}(G_p) \leq \End_{\zz}(G) \leq \prod \End_{\zz}(G_p)$. Suppose that $\frac{G}{\oplus_{p\in \mathbb{P}} G_p}$ is torsion-free divisible. %In this article we characterize pure subrings of $\prod_{p\in \mathbb{P}} \End(G_p)$. We show that $\End_{\zz}(G)$ is a sp-group and every subring $R$ of $\prod \End_{\zz}(G_p)$, containing $\oplus \End_{\zz}(G_p)$ is pure if and only if $R=\mathbb{M}_T=\{x\in \prod_{p\in \mathbb{P}}\End(G_p) \;|\; \exists k\in \nn \;\mbox{\rm{such that}} \;\; kx \in T \},$ where $T$ is a subring of $\prod_{p\in \mathbb{P}}\End(G_p)$. We observe that $\frac{\mathbb{M}_T}{\oplus_{p\in \mathbb{P}}\End(G_p)}$ is (ring) isomorphic with $T\otimes_{\zz} \qq$. Moreover, we conclude that a significant number of the examples around the topic can be easily obtained and described by choosing an appropriate subring $T$. |
| title | On pure subrings of sp-groups |
| topic | Group Theory |
| url | https://arxiv.org/abs/2401.07742 |