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Simple Group Graded Rings and Maximal Commutativity

Öinert, Johan LU (2009) In Preprints in Mathematical Sciences1999-01-01+01:00 2009(6).
Abstract
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... (More)
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. (Less)
Please use this url to cite or link to this publication:
author
organization
publishing date
type
Contribution to journal
publication status
unpublished
subject
keywords
crossed products, Ideals, graded rings, simple rings, maximal commutative subrings, invariant ideals, Picard groups, minimal dynamical systems
in
Preprints in Mathematical Sciences1999-01-01+01:00
volume
2009
issue
6
pages
16 pages
publisher
Lund University
external identifiers
  • other:LUTFMA-5111-2009
ISSN
1403-9338
project
Non-commutative Geometry in Mathematics and Physics
Non-commutative Analysis of Dynamics, Fractals and Wavelets
language
English
LU publication?
yes
id
3b9937ad-852d-40c6-9ebe-298b061e2827 (old id 1370306)
alternative location
http://www.maths.lth.se/matematiklth/personal/oinert/PDF-files/Simple_Group_Graded_Rings_and_Maximal_Commutativity_2009-6.pdf
date added to LUP
2009-05-19 14:16:59
date last changed
2016-11-25 14:10:30
@article{3b9937ad-852d-40c6-9ebe-298b061e2827,
  abstract     = {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.},
  author       = {Öinert, Johan},
  issn         = {1403-9338},
  keyword      = {crossed products,Ideals,graded rings,simple rings,maximal commutative subrings,invariant ideals,Picard groups,minimal dynamical systems},
  language     = {eng},
  number       = {6},
  pages        = {16},
  publisher    = {Lund University},
  series       = {Preprints in Mathematical Sciences1999-01-01+01:00},
  title        = {Simple Group Graded Rings and Maximal Commutativity},
  volume       = {2009},
  year         = {2009},
}