Lecturers and Teaching Assistants
Dr. Josephine Shamash
Course Schedule and Location
Second Semester
Monday, 09:00 - 11:00, Goldsmith, Rm 208
08/04/2024
Field of Study, Course Type and Credit Points
Mathematics and Computer Science: Lecture; Elective; Regular; 2.00 points
Chemical Sciences: Lecture; Elective; 2.00 points
Life Sciences (Brain Sciences: Systems, Computational and Cognitive Neuroscience Track): Lecture; Elective; 2.00 points
Prerequisites
The course is suitable for first-year graduate students in mathematics and computer science who have studied at least 2 years of undergraduate algebra courses.
Attendance and participation
Estimated Weekly Independent Workload (in hours)
Syllabus
This is a basic graduate course in algebra, whose purpose is to provide a good foundation for more advanced and specific algebra courses.
Topics to be covered in the course:
I. Structure theory of modules
- Artinian and Noetherian modules
- Schreier refinement theorem, Jordan-Holder theorem, Krull-Schmidt theorem.
- Completely reducible modules, Schur's lemma.
- Tensor products of modules
- Projective and injective modules
- Wedderburn-Artin theorem for simple rings.
II. Structure theory of rings
- Primitivity and semi-primitivity
- Jacobson radical
- Density theorems
- Artinian rings
- Wedderburn-Artin structure theorems for primitive and semi-primitive artinian rings.
- Commutative artinian rings and the Hilbert Nullstellensatz (if time permits).
Learning Outcomes
Upon successful completion of the course students will be able to:
- Demonstrate comprehension of the structure of modules over rings, in particular the structure of artinian and noetherian modules.
- Apply the theorems and methods of representation theory and modules to obtain insights and understanding of the classical theorems in the structure of rings.
- Demonstrate comprehension of the structure of non-commutative rings, in particular the structure of primitive and semi-primitive artinian rings.
Reading List
- N. Jacobson, Basic Algebra II
- I. M. Isaacs, Algebra: A Graduate Course
- S. Lang, Algebra