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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
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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\to \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
Email: tmarley1@unl.edu

DOI: https://doi.org/10.1090/proc/13287
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

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