  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.
