Ideals and Maximal Commutative Subrings of Graded Rings
Author
Summary, in English
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.
Department/s
Publishing year
2009
Language
English
Publication/Series
Doctoral Theses in Mathematical Sciences
Volume
2009:5
Full text
Document type
Dissertation
Publisher
Centre for Mathematical Sciences, Lund University
Topic
- Mathematics
Keywords
- ideals
- simple rings
- maximal commutativity
- Crossed products
- graded rings
Status
Published
Project
- Non-commutative Analysis of Dynamics, Fractals and Wavelets
- Non-commutative Geometry in Mathematics and Physics
Research group
- Non-commutative Geometry
Supervisor
- Sergei Silvestrov
ISBN/ISSN/Other
- ISSN: 1404-0034
- ISBN: 978-91-628-7832-0
Defence date
17 August 2009
Defence time
13:15
Defence place
Lecture hall MH:C, Centre for Mathematical Sciences, Sölvegatan 18, Lund university, Faculty of Engineering
Opponent
- Søren Eilers (Professor)