Ideals and Maximal Commutative Subrings of Graded Rings
(2009) In Doctoral Theses in Mathematical Sciences 2009:5. Abstract
 This thesis is mainly concerned with the intersection between ideals and (maximal commutative) subrings of graded rings. The motivation for this investigation originates in the theory of C*crossed product algebras associated to topological dynamical systems, where connections between intersection properties of ideals and maximal commutativity of certain subalgebras are wellknown. In the last few years, algebraic analogues of these C*algebra theorems have been proven by C. Svensson, S. Silvestrov and M. de Jeu for different kinds of skew group algebras arising from actions of the group Z. This raised the question whether or not this could be further generalized to other types of (strongly) graded rings. In this thesis we show that it can... (More)
 This thesis is mainly concerned with the intersection between ideals and (maximal commutative) subrings of graded rings. The motivation for this investigation originates in the theory of C*crossed product algebras associated to topological dynamical systems, where connections between intersection properties of ideals and maximal commutativity of certain subalgebras are wellknown. In the last few years, algebraic analogues of these C*algebra theorems have been proven by C. Svensson, S. Silvestrov and M. de Jeu for different kinds of skew group algebras arising from actions of the group Z. This raised the question whether or not this could be further generalized to other types of (strongly) graded rings. In this thesis we show that it can indeed be done for many other types of graded rings and actions!
Given any (category) graded ring, there is a canonical subring which is referred to as the neutral component or the coefficient subring. Through this thesis we successively show that for algebraic crossed products, crystalline graded rings, general strongly graded rings and (under some conditions) groupoid crossed products, each nonzero ideal of the ring has a nonzero intersection with the commutant of the center of the neutral component subring. In particular, if the neutral component subring is maximal commutative in the ring this yields that each nonzero ideal of the ring has a nonzero intersection with the neutral component subring.
Not only are ideal intersection properties interesting in their own right, they also play a key role when investigating simplicity of the ring itself. For strongly group graded rings, there is a canonical action such that the grading group acts as automorphisms of certain subrings of the graded ring. By using the previously mentioned ideal intersection properties we are able to relate Gsimplicity of these subrings to simplicity of the ring itself. It turns out that maximal commutativity of the subrings plays a key role here! Necessary and sufficient conditions for simplicity of a general skew group ring are not known. In this thesis we resolve this problem for skew group rings with commutative coefficient rings. (Less)
Please use this url to cite or link to this publication:
http://lup.lub.lu.se/record/1433647
 author
 Öinert, Johan ^{LU}
 supervisor

 Sergei Silvestrov ^{LU}
 opponent

 Professor Eilers, Søren, University of Copenhagen, Denmark
 organization
 publishing date
 2009
 type
 Thesis
 publication status
 published
 subject
 keywords
 ideals, simple rings, maximal commutativity, Crossed products, graded rings
 in
 Doctoral Theses in Mathematical Sciences
 volume
 2009:5
 pages
 183 pages
 publisher
 Centre for Mathematical Sciences, Lund University
 defense location
 Lecture hall MH:C, Centre for Mathematical Sciences, Sölvegatan 18, Lund university, Faculty of Engineering
 defense date
 20090817 13:15
 ISSN
 14040034
 ISBN
 9789162878320
 project
 Noncommutative Geometry in Mathematics and Physics
 Noncommutative Analysis of Dynamics, Fractals and Wavelets
 language
 English
 LU publication?
 yes
 id
 d4ea49a1d4934a9eaeb6d00799fcd02d (old id 1433647)
 date added to LUP
 20090701 13:35:57
 date last changed
 20160919 08:44:49
@phdthesis{d4ea49a1d4934a9eaeb6d00799fcd02d, abstract = {This thesis is mainly concerned with the intersection between ideals and (maximal commutative) subrings of graded rings. The motivation for this investigation originates in the theory of C*crossed product algebras associated to topological dynamical systems, where connections between intersection properties of ideals and maximal commutativity of certain subalgebras are wellknown. In the last few years, algebraic analogues of these C*algebra theorems have been proven by C. Svensson, S. Silvestrov and M. de Jeu for different kinds of skew group algebras arising from actions of the group Z. This raised the question whether or not this could be further generalized to other types of (strongly) graded rings. In this thesis we show that it can indeed be done for many other types of graded rings and actions!<br/><br> <br/><br> Given any (category) graded ring, there is a canonical subring which is referred to as the neutral component or the coefficient subring. Through this thesis we successively show that for algebraic crossed products, crystalline graded rings, general strongly graded rings and (under some conditions) groupoid crossed products, each nonzero ideal of the ring has a nonzero intersection with the commutant of the center of the neutral component subring. In particular, if the neutral component subring is maximal commutative in the ring this yields that each nonzero ideal of the ring has a nonzero intersection with the neutral component subring.<br/><br> <br/><br> Not only are ideal intersection properties interesting in their own right, they also play a key role when investigating simplicity of the ring itself. For strongly group graded rings, there is a canonical action such that the grading group acts as automorphisms of certain subrings of the graded ring. By using the previously mentioned ideal intersection properties we are able to relate Gsimplicity of these subrings to simplicity of the ring itself. It turns out that maximal commutativity of the subrings plays a key role here! Necessary and sufficient conditions for simplicity of a general skew group ring are not known. In this thesis we resolve this problem for skew group rings with commutative coefficient rings.}, author = {Öinert, Johan}, isbn = {9789162878320}, issn = {14040034}, keyword = {ideals,simple rings,maximal commutativity,Crossed products,graded rings}, language = {eng}, pages = {183}, publisher = {Centre for Mathematical Sciences, Lund University}, school = {Lund University}, series = {Doctoral Theses in Mathematical Sciences}, title = {Ideals and Maximal Commutative Subrings of Graded Rings}, volume = {2009:5}, year = {2009}, }