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- Liberal arts and basic science courses
- Academic unit or major
- Graduate major in Mathematics
- Instructor(s)
- Naito Satoshi
- Class Format
- Lecture
- Media-enhanced courses
- Day/Period(Room No.)
- Tue3-4(H137)
- Group
- -
- Course number
- MTH.A403
- Credits
- 1
- Academic year
- 2016
- Offered quarter
- 3Q
- Syllabus updated
- 2016/12/14
- Lecture notes updated
- -
- Language used
- Japanese
- Access Index
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Course description and aims
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.
Student learning outcomes
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.
Keywords
Modules over a ring, Noetherian modules, Krull-Remak-Schmidt theorem, group representations, complete reducibility
Competencies that will be developed
| ✔ Specialist skills | Intercultural skills | Communication skills | Critical thinking skills | Practical and/or problem-solving skills |
Class flow
Standard lecture course.
Course schedule/Required learning
| Course schedule | Required learning | |
|---|---|---|
| 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 |
Textbook(s)
Toshiyuki Katsura, Algebra II: Modules over a Ring, Toudaishuppan (Japanese)
Reference books, course materials, etc.
Unspecified.
Assessment criteria and methods
Based on the reports with answers of exercise problems presented in the class.
Related courses
- MTH.A404 ： Advanced topics in Algebra D
- MTH.A301 ： Algebra I
- MTH.A302 ： Algebra II
Prerequisites (i.e., required knowledge, skills, courses, etc.)
None required.
