The main subjects of this course are the notion of modules over a ring and some of their basic properties, in particular, basic properties of Noetherian modules.
In this course, we first introduce the notion of modules over a ring, and then explain some of the basic properties of Noetherian modules. Next, we explain the Krull-Remak-Schmidt theorem about the uniqueness of decompositions into indecomposable submodules of a module. Finally, as a typical example of modules over a ring, we explain elementary facts about the representation theory of finite groups. This course will be succeeded by "Advanced courses in Algebra D" in the fourth quarter.
The theory of modules over a ring can be thought of as a generalization and a further development of linear algebra, which is the theory of
vector spaces and linear mappings. Also, the notion of modules over a ring is most fundamental in algebra, and is applicable to describe a wide variety of objects not only in algebra, but also in the whole mathematics. The aim of this course is to make students familiar with this notion and understand some of their basic properties, and enable them to make use of 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 | Basic 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.
Judging from the performance level of the exercises given during the class.
None required.