$\Sigma$-pure injectivity and Brown representability

TL;DR

This paper characterizes $ ext{Sigma}$-pure injective modules via additive and product classes, and links Brown representability in homotopy categories to dual categories, providing new insights into pure-semisimple rings.

Contribution

It establishes a new characterization of $ ext{Sigma}$-pure injective modules and connects Brown representability to dual categories in the context of module theory.

Findings

Characterization of $ ext{Sigma}$-pure injective modules as $ ext{Add}(M) extless extgreater ext{Prod}(M)$

Equivalence of Brown representability in homotopy categories and their duals for unital rings

New characterizations for right pure-semisimple rings

Abstract

We prove that a right $R$-module $M$ is $\Sigma$-pure injective if and only if $\mathrm{Add}(M)\subseteq \mathrm{Prod}(M)$. Consequently, if $R$ is a unital ring, the homotopy category $\mathbf{K}({\mathrm{Mod}\text{-} R})$ satisfies the Brown Representability Theorem if and only if the dual category has the same property. We also apply the main result to provide new characterizations for right pure-semisimple rings or to give a partial positive answer to a question of G. Bergman.