Ideals and Maximal Commutative Subrings of Graded Rings
Författare
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.
Avdelning/ar
Publiceringsår
2009
Språk
Engelska
Publikation/Tidskrift/Serie
Doctoral Theses in Mathematical Sciences
Volym
2009:5
Fulltext
Dokumenttyp
Doktorsavhandling
Förlag
Centre for Mathematical Sciences, Lund University
Ämne
- Mathematics
Nyckelord
- ideals
- simple rings
- maximal commutativity
- Crossed products
- graded rings
Status
Published
Projekt
- Non-commutative Analysis of Dynamics, Fractals and Wavelets
- Non-commutative Geometry in Mathematics and Physics
Forskningsgrupp
- Non-commutative Geometry
Handledare
- Sergei Silvestrov
ISBN/ISSN/Övrigt
- ISSN: 1404-0034
- ISBN: 978-91-628-7832-0
Försvarsdatum
17 augusti 2009
Försvarstid
13:15
Försvarsplats
Lecture hall MH:C, Centre for Mathematical Sciences, Sölvegatan 18, Lund university, Faculty of Engineering
Opponent
- Søren Eilers (Professor)