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 Ggraded ring where R_e is maximal commutative in R, then R is a simple ring if and only if R_e is Gsimple (i.e. there are no nontrivial Ginvariant ideals). We also show that if R_e is commutative (not necessarily maximal commutative) and the commutant of R_e is Gsimple, then R is a simple ring. These results apply to Gcrossed 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 Ggraded ring where R_e is maximal commutative in R, then R is a simple ring if and only if R_e is Gsimple (i.e. there are no nontrivial Ginvariant ideals). We also show that if R_e is commutative (not necessarily maximal commutative) and the commutant of R_e is Gsimple, then R is a simple ring. These results apply to Gcrossed 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 Ggraded 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 unpublished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:LUTFMA51112009
 ISSN
 14039338
 project
 Noncommutative Analysis of Dynamics, Fractals and Wavelets
 Noncommutative Geometry in Mathematics and Physics
 language
 English
 LU publication?
 yes
 id
 3b9937ad852d40c69ebe298b061e2827 (old id 1370306)
 alternative location
 http://www.maths.lth.se/matematiklth/personal/oinert/PDFfiles/Simple_Group_Graded_Rings_and_Maximal_Commutativity_20096.pdf
 date added to LUP
 20160401 14:40:28
 date last changed
 20181121 20:29:00
@article{3b9937ad852d40c69ebe298b061e2827, 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 Ggraded ring where R_e is maximal commutative in R, then R is a simple ring if and only if R_e is Gsimple (i.e. there are no nontrivial Ginvariant ideals). We also show that if R_e is commutative (not necessarily maximal commutative) and the commutant of R_e is Gsimple, then R is a simple ring. These results apply to Gcrossed 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 Ggraded 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 = {14039338}, 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/PDFfiles/Simple_Group_Graded_Rings_and_Maximal_Commutativity_20096.pdf}, volume = {2009}, year = {2009}, }