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Simple group graded rings and maximal commutativity

Öinert, Johan LU (2009) Satellite Conference of the 5th European Congress of Mathematics In Contemporary Mathematics 503. p.159-175
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 = 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... (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 = 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. (Less)
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author
organization
publishing date
type
Chapter in Book/Report/Conference proceeding
publication status
published
subject
keywords
Graded rings, Ideals, Simple rings, Maximal commutative subrings, Picard groups, Invariant ideals, Crossed products, Skew group rings, Minimal dynamical systems
in
Contemporary Mathematics
volume
503
pages
159 - 175
publisher
Amer Mathematical Soc
conference name
Satellite Conference of the 5th European Congress of Mathematics
external identifiers
  • wos:000275898700010
ISSN
0271-4132
1098-3627
ISBN
978-0-8218-4747-3
language
English
LU publication?
yes
id
3f775601-1f4a-40aa-856a-cd014f0a3934 (old id 2536610)
date added to LUP
2012-05-08 13:37:43
date last changed
2016-07-06 13:05:10
@inproceedings{3f775601-1f4a-40aa-856a-cd014f0a3934,
  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 = 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.},
  author       = {Öinert, Johan},
  booktitle    = {Contemporary Mathematics},
  isbn         = {978-0-8218-4747-3},
  issn         = {0271-4132},
  keyword      = {Graded rings,Ideals,Simple rings,Maximal commutative subrings,Picard groups,Invariant ideals,Crossed products,Skew group rings,Minimal dynamical systems},
  language     = {eng},
  pages        = {159--175},
  publisher    = {Amer Mathematical Soc},
  title        = {Simple group graded rings and maximal commutativity},
  volume       = {503},
  year         = {2009},
}