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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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$\Sigma$-pure injectivity and Brown representability
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by Simion Breaz PDF
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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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