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Bimodules in Group Graded Rings
Blekinge Institute of Technology, Faculty of Engineering, Department of Mathematics and Natural Sciences. Blekinge Inst Technol, Dept Math & Nat Sci, S-37179 Karlskrona, Sweden..ORCID iD: 0000-0001-8095-0820
2017 (English)In: Algebras and Representation Theory, ISSN 1386-923X, E-ISSN 1572-9079, Vol. 20, no 6, p. 1483-1494Article in journal (Refereed) Published
Abstract [en]

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 = aS center dot (gaG) 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 G aSc Ha dagger broken vertical bar aS center dot (haH) 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) aS dagger T aS dagger R 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.

Place, publisher, year, edition, pages
SPRINGER , 2017. Vol. 20, no 6, p. 1483-1494
Keywords [en]
Graded ring, Strongly graded ring, Crossed product, Skew group ring, Bimodule, Picard group
National Category
Algebra and Logic
Identifiers
URN: urn:nbn:se:bth-15657DOI: 10.1007/s10468-017-9696-xISI: 000416229800008OAI: oai:DiVA.org:bth-15657DiVA, id: diva2:1166130
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open access

Available from: 2017-12-14 Created: 2017-12-14 Last updated: 2017-12-14

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