$\Sigma$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 puresemisimple rings or to give a partial positive answer to a question of G. Bergman.
 Publication:

arXiv eprints
 Pub Date:
 April 2013
 arXiv:
 arXiv:1304.0979
 Bibcode:
 2013arXiv1304.0979B
 Keywords:

 Mathematics  Rings and Algebras;
 Mathematics  KTheory and Homology;
 Mathematics  Representation Theory
 EPrint:
 6 pages