Rings whose indecomposable modules are pure-projective or pure-injective
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2011
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| _version_ | 1866909676049793024 |
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| author | Couchot, François |
| author_facet | Couchot, François |
| contents | Let $\mathcal{P}$ be the class of rings for which every indecomposable right module is pure-projective or pure-injective. When $R$ is a Noetherian local commutative ring of maximal ideal $P$, it is proven that $R\in\mathcal{P}$ if and only if $R$ is either an artinian valuation ring or a discrete valuation domain of rank one with rank($\widetilde{R}$)$\leq 2$ where $\widetilde{R}$ is the completion of $R$ in its $P$-adic topology. Let $R$ be a commutative ring. Then $R\in\mathcal{P}$ if and only if $R$ is a clean arithmetical ring with $R_P\in\mathcal{P}$ for each maximal ideal $P$ of $R$. Moreover, $R$ is a semi-perfect ring when it is Noetherian. Some examples of commutative rings of the class $\mathcal{P}$ are given. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1108_5707 |
| institution | arXiv |
| publishDate | 2011 |
| record_format | arxiv |
| spellingShingle | Rings whose indecomposable modules are pure-projective or pure-injective Couchot, François Rings and Algebras Let $\mathcal{P}$ be the class of rings for which every indecomposable right module is pure-projective or pure-injective. When $R$ is a Noetherian local commutative ring of maximal ideal $P$, it is proven that $R\in\mathcal{P}$ if and only if $R$ is either an artinian valuation ring or a discrete valuation domain of rank one with rank($\widetilde{R}$)$\leq 2$ where $\widetilde{R}$ is the completion of $R$ in its $P$-adic topology. Let $R$ be a commutative ring. Then $R\in\mathcal{P}$ if and only if $R$ is a clean arithmetical ring with $R_P\in\mathcal{P}$ for each maximal ideal $P$ of $R$. Moreover, $R$ is a semi-perfect ring when it is Noetherian. Some examples of commutative rings of the class $\mathcal{P}$ are given. |
| title | Rings whose indecomposable modules are pure-projective or pure-injective |
| topic | Rings and Algebras |
| url | https://arxiv.org/abs/1108.5707 |