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Injective modules under faithfully flat ring extensions

Authors: Lars Winther Christensen and Fatih Köksal
Journal: Proc. Amer. Math. Soc. 144 (2016), 1015-1020
MSC (2010): Primary 13C11; Secondary 13D05
Published electronically: July 30, 2015
MathSciNet review: 3447655
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Abstract: Let $ R$ be a commutative ring and let $ S$ be an $ R$-algebra. It is well-known that if $ N$ is an injective $ R$-module, then $ \operatorname {Hom}_R(S,N)$ is an injective $ S$-module. The converse is not true, not even if $ R$ is a commutative noetherian local ring and $ S$ is its completion, but it is close: It is a special case of our main theorem that, in this setting, an $ R$-module $ N$ with $ \operatorname {Ext}^{>0}_R(S,N) =0$ is injective if $ \operatorname {Hom}_R(S,N)$ is an injective $ S$-module.

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Additional Information

Lars Winther Christensen
Affiliation: Department of Mathematics, Texas Tech University, Lubbock, Texas 79409

Fatih Köksal
Affiliation: Department of Mathematics, Texas Tech University, Lubbock, Texas 79409
Address at time of publication: Department of Mathematics and Computer Science, Lewis University, One Univeristy Parkway, Romeoville, Illinois 60446-2200

Keywords: Injective module, faithfully flat ring extension, co-base change
Received by editor(s): September 29, 2014
Received by editor(s) in revised form: March 24, 2015
Published electronically: July 30, 2015
Additional Notes: This research was partly supported by a Simons Foundation Collaboration Grant for Mathematicians, award no. 281886, and by grant no. H98230-14-0140 from the National Security Agency.
Communicated by: Irena Peeva
Article copyright: © Copyright 2015 American Mathematical Society

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