$Σ$-pure injectivity and Brown representability

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.