## $\Sigma$-pure injectivity and Brown representability

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## 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.## References

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## Additional Information

**Simion Breaz**- Affiliation: Faculty of Mathematics and Computer Science, “Babeş-Bolyai” University, Str. Mihail Kogălniceanu 1, 400084 Cluj-Napoca, Romania
- Email: bodo@math.ubbcluj.ro
- Received by editor(s): March 25, 2013
- Received by editor(s) in revised form: April 18, 2013, July 25, 2013, and February 7, 2014
- Published electronically: January 22, 2015
- Additional Notes: The author’s research was supported by the CNCS-UEFISCDI grant PN-II-RU-TE-2011-3-0065
- Communicated by: Birge Huisgen-Zimmermann
- © Copyright 2015 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**143**(2015), 2789-2794 - MSC (2010): Primary 16D90, 18G35
- DOI: https://doi.org/10.1090/S0002-9939-2015-12481-1
- MathSciNet review: 3336604