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

TL;DR

This paper characterizes rings where every indecomposable module is pure-projective or pure-injective, providing specific classifications for Noetherian local rings and general commutative rings, with examples included.

Contribution

It offers a complete classification of rings in class , especially for Noetherian local and arithmetical rings, and explores their properties and examples.

Findings

Noetherian local rings in  are either artinian valuation rings or discrete valuation domains with certain completion properties.

A commutative ring belongs to  if and only if it is a clean arithmetical ring with localizations in .

Noetherian rings in  are semi-perfect.

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.