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A change of rings result for Matlis reflexivity


Authors: Douglas J. Dailey and Thomas Marley
Journal: Proc. Amer. Math. Soc. 145 (2017), 1837-1841
MSC (2010): Primary 13C05; Secondary 13C13
DOI: https://doi.org/10.1090/proc/13287
Published electronically: January 26, 2017
MathSciNet review: 3611300
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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

Keywords: Matlis reflexive, minimal injective cogenerator
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
Article copyright: © Copyright 2017 American Mathematical Society