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$\Sigma$-pure injectivity and Brown representability

Author: Simion Breaz
Journal: Proc. Amer. Math. Soc. 143 (2015), 2789-2794
MSC (2010): Primary 16D90, 18G35
Published electronically: January 22, 2015
MathSciNet review: 3336604
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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.

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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

Keywords: $\Sigma$-pure injective module, pure-semisimple ring, $p$-functor, Brown representability theorem, homotopy category
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
Article copyright: © Copyright 2015 American Mathematical Society