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Ideals and Maximal Commutative Subrings of Graded Rings

Öinert, Johan LU (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 well-known. 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 well-known. 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 G-simplicity 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:
author
supervisor
oppenent
  • Professor Søren Eilers, University of Copenhagen, Denmark
organization
year
type
Dissertation (Composite)
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
defense location
Lecture hall MH:C, Centre for Mathematical Sciences, Sölvegatan 18, Lund university, Faculty of Engineering
defense date
2009-08-17 13:15
ISSN
1404-0034
ISBN
978-91-628-7832-0
project
Non-commutative Analysis of Dynamics, Fractals and Wavelets
project
Non-commutative Geometry in Mathematics and Physics
research group
Non-commutative Geometry
funder
LieGrits EC Marie Curie Research Training Network
funder
Swedish Research Council
funder
STINT
funder
Kungliga fysiografiska sällskapet
funder
The Royal Swedish Academy of Sciences
funder
Crafoord Foundation
funder
LU
language
English
LU publication?
yes
id
1433647
date added to LUP
2009-07-01 13:35:57
date last changed
2009-07-01 13:35:57
@phdthesis{1433647,
  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 well-known. 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 G-simplicity 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       = {\"O inert, Johan},
  isbn         = {978-91-628-7832-0},
  issn         = {1404-0034},
  keyword      = {ideals,simple rings,maximal commutativity,Crossed products,graded rings},
  language     = {eng},
  pages        = {183},
  publisher    = {Centre for Mathematical Sciences},
  school       = {Lund University},
  series       = {Doctoral Theses in Mathematical Sciences},
  title        = {Ideals and Maximal Commutative Subrings of Graded Rings},
  volume       = {2009:5},
  year         = {2009},
}