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- Academic unit or major
- Mathematics
- Instructor(s)
- Kato Fumiharu Wakabayashi Yasuhiro
- Class Format
- Lecture (Face-to-face)
- Media-enhanced courses
- Day/Period(Room No.)
- Wed3-4(H103) Thr3-4(H103)
- Group
- -
- Course number
- ZUA.A301
- Credits
- 2
- Academic year
- 2022
- Offered quarter
- 1-2Q
- Syllabus updated
- 2022/3/16
- Lecture notes updated
- -
- Language used
- Japanese
- Access Index
-
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. The second half of the course will be devoted on basic topics surrounding Noetherian and Artinian rings, local rings, and homological algebras. 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).
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.
4) Understand the notion of Noethrian and Artinian rings, and make use of fundamental operations for them correctly.
5) Understand the notion of local rings, and make use of fundamental operations for them correctly.
6) Understand homological algebra, and make use of fundamental operations for them correctly.
Keywords
rings, ideal, residue rings, modules, tensor products, localization, Noetherian rings, Artinian rings, local rings, homological algebra
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 | Commutative rings and ideals | Details will be provided during each class session. |
| Class 2 | Modules over a ring (1) | Details will be provided during each class session. |
| Class 3 | Modules over a ring (2) | Details will be provided during each class session. |
| Class 4 | Modules over a ring (3) | Details will be provided during each class session. |
| Class 5 | Tensor products of modules (1) | Details will be provided during each class session. |
| Class 6 | Tensor products of modules (2) | Details will be provided during each class session. |
| Class 7 | Localization | Details will be provided during each class session. |
| Class 8 | Noetherian rings and Artinian rings (1) | Details will be provided during each class session. |
| Class 9 | Noetherian rings and Artinian rings (2) | Details will be provided during each class session. |
| Class 10 | Local rings | Details will be provided during each class session. |
| Class 11 | Homological algebra (1) | Details will be provided during each class session. |
| Class 12 | Homological algebra (2) | Details will be provided during each class session. |
| Class 13 | Homological algebra (3) | Details will be provided during each class session. |
| Class 14 | Advanced topics | 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.A301 ： Algebra I
- MTH.A302 ： Algebra II
- MTH.A201 ： Introduction to Algebra I
- MTH.A202 ： Introduction to Algebra II
- ZUA.A302 ： Exercises in Algebra B I
Prerequisites (i.e., required knowledge, skills, courses, etc.)
Students must 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.
Students are strongly recommended to take ZUA.A302: Exercises in Algebra B I (if not passed yet) at the same time.
Other
None in particular.
