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arXiv · 1108.5707

Rings whose indecomposable modules are pure-projective or pure-injective

François Couchot
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Abstract

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.

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François Couchot. 2011-08-29. Rings whose indecomposable modules are pure-projective or pure-injective. https://arxiv.org/abs/1108.5707

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