The main topics of this course are the concept of modules over rings and their properties, in particular the properties of Noetherian modules. This course will first cover the basic concepts of the theory of modules over rings, then go over the properties of Noetherian modules. Next, the instructor explains Krull-Remak-Schmidt's theorem related to the uniqueness of the indecomposable decomposition of modules over rings. The instructor then goes over introductory topics for representation theory of finite groups as a typical example of modules over rings. This course is followed by Advanced Topics in Algebra D.
The theory of modules over rings expands and develops for more general cases the vector spaces and linear mapping theories learned in linear algebra. These basic concepts of algebra are not restricted to algebra, but rather can be broadly applied to mathematics as a whole. The purpose of this course is for students to become familiar with these concepts, to clearly understand their basic properties, and to be able to use them correctly.
By the end of this course, students will be able to:
1) Explain the definition and some of the basic properties of modules over a ring.
2) Understand some of the basic properties of Noetherian modules.
3) Make use of the Krull-Remak-Schmidt theorem correctly.
4) Understand the elementary facts about the representation theory of finite groups.
Modules over a ring, Noetherian modules, Krull-Remak-Schmidt theorem, group representations, complete reducibility
Intercultural skills | Communication skills | Specialist skills | Critical thinking skills | Practical and/or problem-solving skills |
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- | - | ✔ | - | - |
Standard lecture course.
Course schedule | Required learning | |
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Class 1 | Definition of modules over a ring | Details will be provided during each class session |
Class 2 | Submodules and homomorphisms | Details will be provided during each class session |
Class 3 | Direct sums and free modules | Details will be provided during each class session |
Class 4 | Composition series of modules over a ring | Details will be provided during each class session |
Class 5 | Basics facts about Noetherian modules | Details will be provided during each class session |
Class 6 | Krull-Remak-Schmidt theorem | Details will be provided during each class session |
Class 7 | Group representations | Details will be provided during each class session |
Class 8 | Complete reducibility of group representations | Details will be provided during each class session |
Toshiyuki Katsura, Algebra II: Modules over a Ring, Toudaishuppan (Japanese)
Unspecified.
Based on the reports with answers of exercise problems presented in the class.
None required.