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

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.