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- Academic unit or major
- Undergraduate major in Mathematics
- Instructor(s)
- Kato Fumiharu Wakabayashi Yasuhiro
- Class Format
- Lecture / Exercise (Face-to-face)
- Media-enhanced courses
- Day/Period(Room No.)
- Thr3-6(H103)
- Group
- -
- Course number
- MTH.A301
- Credits
- 2
- Academic year
- 2022
- Offered quarter
- 1Q
- Syllabus updated
- 2022/3/16
- Lecture notes updated
- -
- Language used
- Japanese
- Access Index
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Course description and aims
The main topics of this course are the basic concepts and properties related to (commutative) rings, ideals, and modules over them. After reviewing some of the basics on (commutative) rings, ideals, and residue rings mod out by ideals, students will learn the concept of modules over a ring systematically, together with many of the related concepts including submodules, residue modules, linear mappings, homomorphism theorem, direct sums and direct products, exact sequences, Hom modules, free modules, etc. This will be followed by a study on basic topics on tensor products of modules, right exactness of tensor products, and further related concepts (e.g., flatness). This course is followed by "Algebra II."
Rings, ideals, and modules over rings are among the most basic concepts in advanced algebra, which admits wide applications. However, its abstractness would cause several difficulties for newcomers. Students in this course will attempt to solidify these concepts in their mind by becoming familiar with these kinds of abstract concepts through rational integer rings and polynomial rings which are typical examples of (commutative) rings.
Student learning outcomes
By the end of this course, students will be able to:
1) Understand the notions of (commutative) rings and modules over rings.
2) Understand tensor products and make use of them correctly.
3) Understand localization and make use of them correctly.
Keywords
rings, ideal, residue rings, modules, tensor products, localization
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 accompanied by discussion session.
Course schedule/Required learning
| Course schedule | Required learning | |
|---|---|---|
| Class 1 | Commutative rings and ideals | Details will be provided during each class session. |
| Class 2 | discussion session | Details will be provided during each class session. |
| Class 3 | Modules over a ring (1) | Details will be provided during each class session. |
| Class 4 | discussion session | Details will be provided during each class session. |
| Class 5 | Modules over a ring (2) | Details will be provided during each class session. |
| Class 6 | discussion session | Details will be provided during each class session. |
| Class 7 | Modules over a ring (3) | Details will be provided during each class session. |
| Class 8 | discussion session | |
| Class 9 | Tensor products of modules (1) | Details will be provided during each class session. |
| Class 10 | discussion session | Details will be provided during each class session. |
| Class 11 | Tensor products of modules (2) | Details will be provided during each class session. |
| Class 12 | discussion session | Details will be provided during each class session. |
| Class 13 | Localization | Details will be provided during each class session. |
| Class 14 | discussion session | Details will be provided during each class session. |
Out-of-Class Study Time (Preparation and Review)
To enhance effective learning, students are encouraged to spend approximately 100 minutes preparing for class and another 100 minutes reviewing class content afterwards (including assignments) for each class.
They should do so by referring to textbooks and other course material.
Textbook(s)
TBA
Reference books, course materials, etc.
TBA
Assessment criteria and methods
TBA
Related courses
- MTH.A201 ： Introduction to Algebra I
- MTH.A202 ： Introduction to Algebra II
- MTH.A203 ： Introduction to Algebra III
- MTH.A204 ： Introduction to Algebra IV
- MTH.A302 ： Algebra II
Prerequisites (i.e., required knowledge, skills, courses, etc.)
Students are required to have successfully completed Linear Algebra I/Recitation, Linear Algebra II, Linear Algebra Recitation II, Advanced Linear Algebra I, II, and Introduction to Algebra I, II, III, IV; or, they must have equivalent knowledge.
Other
None in particular.
