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Simple group graded rings and maximal commutativity

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Author

  • Johan Öinert

Summary, in English

In this paper we provide necessary and sufficient conditions for strongly group graded rings to be simple. For a strongly group graded ring R = circle plus(g is an element of G)R(g) the grading group G acts, in a natural way, as automorphisms of the comrnutant of the neutral component subring R(e) in R and of the center of R(e). We show that if R. is a strongly G-graded ring where R(e), is maximal commutative in R(e), then R is a simple ring if and only if R(e), is G-simple (i.e. there are no nontrivial G-invariant ideals). We also show that if R(e), is commutative (not necessarily maximal commutative) and the commutant of R(e)., is G-simple, then R. is a simple ring. These results apply to G-crossed products in particular. A skew group ring R(e), G, where R(e), is commutative, is shown to be a simple ring if and only if R(e), is G-simple and maximal commutative in R(e), >(sigma), G. As an interesting example we consider the skew group algebra C(X) (sic) ((h) over bar) Z associated to a topological dynamical system (X, h). We obtain necessary and sufficient conditions for simplicity of C(X) (sic) ((h) over bar) Z with respect to the dynamics of the dynamical system (X, h), but also with respect to algebraic properties of C(X) as a subalgebra of C(X) (sic) ((h) over bar) Z. Furthermore, we show that for any strongly G-graded ring R each nonzero ideal of R has a nonzero intersection with the commutant of the center of the neutral component.

Publishing year

2009

Language

English

Pages

159-175

Publication/Series

Contemporary Mathematics

Volume

503

Document type

Conference paper

Publisher

American Mathematical Society (AMS)

Topic

  • Mathematics

Keywords

  • Graded rings
  • Ideals
  • Simple rings
  • Maximal commutative subrings
  • Picard groups
  • Invariant ideals
  • Crossed products
  • Skew group rings
  • Minimal dynamical systems

Conference name

Satellite Conference of the 5th European Congress of Mathematics

Conference date

2008-07-21 - 2008-07-25

Conference place

Leiden, Netherlands

Status

Published

ISBN/ISSN/Other

  • ISSN: 0271-4132
  • ISSN: 1098-3627
  • ISBN: 978-0-8218-4747-3