### 2016　Advanced topics in Algebra C

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Instructor(s)
Naito Satoshi
Course component(s)
Lecture
Day/Period(Room No.)
Tue3-4(H137)
Group
-
Course number
MTH.A403
Credits
1
2016
Offered quarter
3Q
Syllabus updated
2016/12/14
Lecture notes updated
-
Language used
Japanese
Access Index

### 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

Intercultural skills Communication skills Specialist 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)

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

None required.