On $S$-injective modules
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| Acceso en línea: | |
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| _version_ | 1866929530543800320 |
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| author | Bennis, Driss Bouziri, Ayoub |
| author_facet | Bennis, Driss Bouziri, Ayoub |
| contents | Let $R$ be a commutative ring with identity, and let $S$ be a multiplicative subset of $R$. In this paper, we introduce the notion of $S$-injective modules as a weak version of injective modules. Among other results, we provide an $S$-version of Baer's characterization of injective modules. We also present an $S$-version of Lambek's characterization of flat modules: an $R$-module $M$ is $S$-flat if and only if its character, $\text{Hom}_{\mathbb{Z}}(M, \mathbb{Q}/\mathbb{Z})$, is an $S$-injective $R$-module. As applications, we establish, under certain conditions, $S$-counterparts of the Cartan--Eilenberg-Bass and Cheatham--Stone characterizations of Noetherian rings. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_02041 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On $S$-injective modules Bennis, Driss Bouziri, Ayoub Commutative Algebra 13C11, 13C13, 13E99 Let $R$ be a commutative ring with identity, and let $S$ be a multiplicative subset of $R$. In this paper, we introduce the notion of $S$-injective modules as a weak version of injective modules. Among other results, we provide an $S$-version of Baer's characterization of injective modules. We also present an $S$-version of Lambek's characterization of flat modules: an $R$-module $M$ is $S$-flat if and only if its character, $\text{Hom}_{\mathbb{Z}}(M, \mathbb{Q}/\mathbb{Z})$, is an $S$-injective $R$-module. As applications, we establish, under certain conditions, $S$-counterparts of the Cartan--Eilenberg-Bass and Cheatham--Stone characterizations of Noetherian rings. |
| title | On $S$-injective modules |
| topic | Commutative Algebra 13C11, 13C13, 13E99 |
| url | https://arxiv.org/abs/2406.02041 |