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|Title||Ideals and Maximal Commutative Subrings of Graded Rings|
Mathematics (Faculty of Technology)
|Full-text||Available as PDF|
|Defence place||Lecture hall MH:C, Centre for Mathematical Sciences, Sölvegatan 18, Lund university, Faculty of Engineering|
|Opponent||Professor Søren Eilers|
|Publication/Series||Doctoral Theses in Mathematical Sciences|
|Publisher||Centre for Mathematical Sciences|
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.
Mathematics and Statistics
|Keywords||simple rings, ideals, maximal commutativity, Crossed products, graded rings|
|Research group||Non-commutative Geometry|
|Project||Non-commutative Analysis of Dynamics, Fractals and Wavelets|
|Project||Non-commutative Geometry in Mathematics and Physics|
LieGrits EC Marie Curie Research Training Network
Swedish Research Council
Kungliga fysiografiska sällskapet
The Royal Swedish Academy of Sciences