C2.5 Non-Commutative Rings (2024-25)

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  Course Materials

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  • Select activity Lecture Notes

    

    Lecture Notes File

    Lectures 1-2 contain an overview of the course, the definitions of Noetherian and Artin rings, and examples: Weyl algebras, group algebras and universal enveloping algebras

    Lectures 3-4 cover polycyclic groups, sufficient conditions for a ring to be Noetherian, filtered and graded rings.

    Lectures 5-7 cover the theory of Artinian rings: Jacobson radical, left-primitive rings and modules, and the Artin- Wedderburn theorem. 

    Lectures 8 and 9 cover non-commutative localisation: the Ore condition, left localisable sets, Goldie rings.

    Lectures 10 and 11 cover dimension theory of modules and Krull dimension of graded modules when the graded ring is commutative.

    Lectures 12-13 cover Rees rings and good filtrations of modules.

    Lectures 14-15 cover the Poisson bracket of the graded ring of the Weyl algebra and a proof of Bernsteins inequality.

    Lecture 16 gives the reduction of Gabber's theorem to the local case. 

  • Select activity Problem Sheet 0

    

    Problem Sheet 0 File
  • Select activity Sheet 1

    

    Sheet 1 Assignment

    Due: Friday, 31 January 2025, 11:00 AM

    This problem sheet is based on lectures in weeks 1 and 2: Group rings, Weyl algebra and Noetherian rings.
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    Sheet 2 Assignment

    This problem sheet is based on lectures from weeks 3 and 4, the structure theory of Artinian rings.
  • Select activity Sheet 3

    

    Sheet 3 Assignment

    Due: Friday, 28 February 2025, 11:00 AM

    This problem sheet is based on material in lectures from weeks 5 and 6: Ore rings and basic properies of dimension of modules.
  • Select activity Sheet 4

    

    Sheet 4 Assignment

    This problem sheet covers material from lectures in weeks 6,7 and 8:
    Rees rings, good filtration, Poisson bracket and Bernstein's inequality.
• Select section Intercollegiate Classes

  Intercollegiate Classes

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    Class Signup Class registration

    Registration start: Monday, 13 January 2025, 12:00 PM

    Registration end: Friday, 14 February 2025, 12:00 PM
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    Class Tutor's Comments Assignment

    Class tutors will use this activity to provide overall feedback to students at the end of the course.
    

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