Simple group graded rings and maximal commutativity
(2009) Satellite Conference of the 5th European Congress of 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)
Please use this url to cite or link to this publication:
https://lup.lub.lu.se/record/2536610
- author
- Öinert, Johan LU
- organization
- publishing date
- 2009
- 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
- host publication
- Contemporary Mathematics
- volume
- 503
- pages
- 159 - 175
- publisher
- American Mathematical Society (AMS)
- conference name
- Satellite Conference of the 5th European Congress of Mathematics
- conference location
- Leiden, Netherlands
- conference dates
- 2008-07-21 - 2008-07-25
- external identifiers
-
- wos:000275898700010
- ISSN
- 1098-3627
- 0271-4132
- ISBN
- 978-0-8218-4747-3
- language
- English
- LU publication?
- yes
- id
- 3f775601-1f4a-40aa-856a-cd014f0a3934 (old id 2536610)
- date added to LUP
- 2016-04-01 12:19:02
- date last changed
- 2019-10-21 13:54:44
@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 = {{1098-3627}},
keywords = {{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 = {{American Mathematical Society (AMS)}},
title = {{Simple group graded rings and maximal commutativity}},
volume = {{503}},
year = {{2009}},
}