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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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A change of rings result for Matlis reflexivity
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by Douglas J. Dailey and Thomas Marley PDF
Proc. Amer. Math. Soc. 145 (2017), 1837-1841 Request permission

Abstract:

Let $R$ be a commutative Noetherian ring and $E$ the minimal injective cogenerator of the category of $R$-modules. An $R$-module $M$ is (Matlis) reflexive if the natural evaluation map $M{\longrightarrow }\mathrm {Hom}_R(\mathrm {Hom}_R(M,E),E)$ is an isomorphism. We prove that if $S$ is a multiplicatively closed subset of $R$ and $M$ is an $R_S$-module which is reflexive as an $R$-module, then $M$ is a reflexive $R_S$-module. The converse holds when $S$ is the complement of the union of finitely many nonminimal primes of $R$, but fails in general.
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Additional Information
  • Douglas J. Dailey
  • Affiliation: Department of Mathematics, University of Dallas, Irving, Texas 75062-4736
  • Email: ddailey@udallas.edu
  • Thomas Marley
  • Affiliation: Department of Mathematics, University of Nebraska-Lincoln, Lincoln, Nebraska 68588-0130
  • MR Author ID: 263869
  • Email: tmarley1@unl.edu
  • Received by editor(s): October 14, 2015
  • Received by editor(s) in revised form: May 11, 2016
  • Published electronically: January 26, 2017
  • Additional Notes: The first author was partially supported by U.S. Department of Education grant P00A120068 (GAANN)
  • Communicated by: Irena Peeva
  • © Copyright 2017 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 145 (2017), 1837-1841
  • MSC (2010): Primary 13C05; Secondary 13C13
  • DOI: https://doi.org/10.1090/proc/13287
  • MathSciNet review: 3611300