Simple Group Graded Rings and Maximal Commutativity
(2009) In Preprints in Mathematical Sciences 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:
https://lup.lub.lu.se/record/1370306
- author
- Öinert, Johan LU
- organization
- publishing date
- 2009
- 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- unpublished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 Sciences
- volume
- 2009
- issue
- 6
- pages
- 16 pages
- publisher
- Lund University
- external identifiers
-
- other:LUTFMA-5111-2009
- ISSN
- 1403-9338
- project
- Non-commutative Analysis of Dynamics, Fractals and Wavelets
- Non-commutative Geometry in Mathematics and Physics
- 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
- 2016-04-01 14:40:28
- date last changed
- 2018-11-21 20:29:00
@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}}, keywords = {{crossed products; Ideals; graded rings; simple rings; maximal commutative subrings; invariant ideals; Picard groups; minimal dynamical systems}}, language = {{eng}}, number = {{6}}, publisher = {{Lund University}}, series = {{Preprints in Mathematical Sciences}}, title = {{Simple Group Graded Rings and Maximal Commutativity}}, url = {{http://www.maths.lth.se/matematiklth/personal/oinert/PDF-files/Simple_Group_Graded_Rings_and_Maximal_Commutativity_2009-6.pdf}}, volume = {{2009}}, year = {{2009}}, }