Simple group graded rings and maximal commutativity
(2009) Satellite Conference of the 5th European Congress of Mathematics 503. p.159175 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 Ggraded ring where R(e), is maximal commutative in R(e), 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. 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 Ggraded ring where R(e), is maximal commutative in R(e), 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. 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 Gsimple 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 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/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
 20080721  20080725
 external identifiers

 wos:000275898700010
 ISSN
 02714132
 10983627
 ISBN
 9780821847473
 language
 English
 LU publication?
 yes
 id
 3f7756011f4a40aa856acd014f0a3934 (old id 2536610)
 date added to LUP
 20160401 12:19:02
 date last changed
 20191021 13:54:44
@inproceedings{3f7756011f4a40aa856acd014f0a3934, 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 Ggraded ring where R(e), is maximal commutative in R(e), 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. 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 Gsimple 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 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}, booktitle = {Contemporary Mathematics}, isbn = {9780821847473}, issn = {02714132}, language = {eng}, pages = {159175}, publisher = {American Mathematical Society (AMS)}, title = {Simple group graded rings and maximal commutativity}, volume = {503}, year = {2009}, }