S-flat cotorsion pair
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866912007793410048 |
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| author | Bennis, Driss Bouziri, Ayoub |
| author_facet | Bennis, Driss Bouziri, Ayoub |
| contents | Let $R$ be a commutative ring, and let $S$ be a multiplicative subset of $R$. In this paper, we investigate the notion of $S$-cotorsion modules. An $R$-module $C$ is called $S$-cotorsion if $\text{Ext}^{1}_{R}(F,C) = 0$ for every $S$-flat $R$-module $F$. Among other results, we establish that the pair $(S\mathcal{F}, S\mathcal{C})$, where $S\mathcal{F}$ denotes the class of all $S$-flat $R$-modules and $S\mathcal{C}$ denotes the class of all $S$-cotorsion modules, forms a hereditary perfect cotorsion pair. As applications, we provide characterizations of $S$-perfect rings in terms of $S$-cotorsion modules. We conclude the paper with results on $S\mathcal{F}$-preenvelopes. Namely, we prove that if every module has an $S\mathcal{F}$-preenvelope, then $R$ is $S$-coherent. Furthermore, we establish the converse under the condition that $R_S$ is a finitely presented $R$-module. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_09242 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | S-flat cotorsion pair Bennis, Driss Bouziri, Ayoub Commutative Algebra Let $R$ be a commutative ring, and let $S$ be a multiplicative subset of $R$. In this paper, we investigate the notion of $S$-cotorsion modules. An $R$-module $C$ is called $S$-cotorsion if $\text{Ext}^{1}_{R}(F,C) = 0$ for every $S$-flat $R$-module $F$. Among other results, we establish that the pair $(S\mathcal{F}, S\mathcal{C})$, where $S\mathcal{F}$ denotes the class of all $S$-flat $R$-modules and $S\mathcal{C}$ denotes the class of all $S$-cotorsion modules, forms a hereditary perfect cotorsion pair. As applications, we provide characterizations of $S$-perfect rings in terms of $S$-cotorsion modules. We conclude the paper with results on $S\mathcal{F}$-preenvelopes. Namely, we prove that if every module has an $S\mathcal{F}$-preenvelope, then $R$ is $S$-coherent. Furthermore, we establish the converse under the condition that $R_S$ is a finitely presented $R$-module. |
| title | S-flat cotorsion pair |
| topic | Commutative Algebra |
| url | https://arxiv.org/abs/2403.09242 |