Σ-pure injectivity and Brown representability

Breaz, Simion

doi:10.1090/S0002-9939-2015-12481-1

$\Sigma$-pure injectivity and Brown representability
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Proc. Amer. Math. Soc. 143 (2015), 2789-2794 Request permission

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.

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