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Algebras and Representation Theory

, Volume 20, Issue 6, pp 1483–1494 | Cite as

Bimodules in Group Graded Rings

  • Johan Öinert
Open Access
Article
  • 159 Downloads

Abstract

In this article we introduce the notion of a controlled group graded ring. Let G be a group, with identity element e, and let R = ⊕ gG R g be a unital G-graded ring. We say that R is G-controlled if there is a one-to-one correspondence between subsets of the group G and (mutually non-isomorphic) R e -sub-bimodules of R, given by GH↦ ⊕ hH R h . For strongly G-graded rings, the property of being G-controlled is stronger than that of being simple. We provide necessary and sufficient conditions for a general G-graded ring to be G-controlled. We also give a characterization of strongly G-graded rings which are G-controlled. As an application of our main results we give a description of all intermediate subrings T with R e TR of a G-controlled strongly G-graded ring R. Our results generalize results for artinian skew group rings which were shown by Azumaya 70 years ago. In the special case of skew group rings we obtain an algebraic analogue of a recent result by Cameron and Smith on bimodules in crossed products of von Neumann algebras.

Keywords

Graded ring Strongly graded ring Crossed product Skew group ring Bimodule Picard group 

Mathematics Subject Classification (2010)

16S35 16W50 16D40 

Notes

Acknowledgements

The author would like to thank an anonymous referee for valuable comments on the manuscript and for kindly having provided Example 1.

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Copyright information

© The Author(s) 2017

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Department of Mathematics and Natural SciencesBlekinge Institute of TechnologyKarlskronaSweden

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