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Simple Group Graded Rings and Maximal Commutativity

På svenska

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 the grading group G acts, in a natural way, as automorphisms of the commutant 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, 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. As an interesting example we consider the skew group algebra C(X) ⋊˜h Z associated to a topological dynamical system (X, h). We obtain necessary and sufficient conditions for simplicity of C(X) ⋊˜h Z with respect to the dynamics of the dynamical system (X, h), but also with respect to algebraic properties of C(X) ⋊˜h 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

Publication/Series

Preprints in Mathematical Sciences

Volume

2009

Issue

6

Document type

Journal article

Publisher

Lund University

Topic

  • Mathematics

Keywords

  • crossed products
  • Ideals
  • graded rings
  • simple rings
  • maximal commutative subrings
  • invariant ideals
  • Picard groups
  • minimal dynamical systems

Status

Unpublished

Project

  • Non-commutative Analysis of Dynamics, Fractals and Wavelets
  • Non-commutative Geometry in Mathematics and Physics

Research group

  • Non-commutative Geometry

ISBN/ISSN/Other

  • ISSN: 1403-9338
  • LUTFMA-5111-2009