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

$\Sigma$-pure injectivity and Brown representability