V0135400
2
A ring is an algebraic structure that generalizes integers, complex numbers, matrices, polynomials, etc. A module over a ring
can be thought of as a vector space whose coefficients are extended to the ring. This course lectures basics on rings and
modules. In the first half part of the course, fundamental notions in the ring theory (ideals, homomorphisms, pincipal ideal
domains, unique factorization domains, etc) are introduced. In the latter half, basics on modules and elementary divisors
are introduced. As applications, the fundamental theorem of finitely generated abelian groups is given. The existence of the
Jordan canonical form is given from a viewpoint that is different to the course "Linear Space".
Understand basics on rings and modules.
- Understand and be able to explain basics on rings and modules.
- Understand and be able to explain basics on the division relation in a ring.
- Explicitly compute elementary divisors of a matrix over a Euclidean domain.
Relationship between 'Goals and Objectives' and 'Course Outcomes'
|
Homework |
Midterm exam |
Final exam |
Total. |
| 1. |
15% |
15% |
0% |
30% |
| 2. |
15% |
10% |
10% |
35% |
| 3. |
20% |
|
15% |
35% |
| Total. |
50% |
25% |
25% |
- |
|
Class schedule |
HW assignments (Including preparation and review of the class.) |
Amount of Time Required |
| 1. |
Rings and examples |
Review Algebra I |
180minutes |
| 2. |
Ideals and the division relation |
Review the last lecture |
180minutes |
| 3. |
Ideals and quotient rings |
Review the last lecture |
180minutes |
| 4. |
The isomorphism theorem |
Review the last lecture |
180minutes |
| 5. |
Unique factorization domain (UFD) |
Review the last lecture |
180minutes |
| 6. |
Principal ideal domain (PID) |
Review the last lecture |
180minutes |
| 7. |
Midterm exam and review |
Review the content of this course |
270minutes |
| 8. |
Modules over a ring |
Review the notion of linear space |
180minutes |
| 9. |
Basis for a module |
Review the last lecture |
180minutes |
| 10. |
Matrix expression |
Review the last lecture |
180minutes |
| 11. |
Elementary divisor theory (1) Introduction from linear algebra |
Review the last lecture |
180minutes |
| 12. |
Elementary divisor theory (2) The case of Euclidean domains |
Review the last lecture |
180minutes |
| 13. |
Finitely generated modules over a PID |
Review the last lecture |
180minutes |
| 14. |
Final exam and review |
Review the content of this course |
270minutes |
| Total. |
- |
- |
2700minutes |
Evaluation method and criteria
Evaluated as indicated in "Course Outcomes" section. A score of 60 or more out of 100 points is required to pass this course.
To pass this course, students should understand basics on rings and modules.
Feedback on exams, assignments, etc.
| ways of feedback |
specific contents about "Other" |
| Feedback in the class |
|
Textbooks and reference materials
None
Linear algebra I, II and Algebra I is assumed.
Office hours and How to contact professors for questions
- 12:30-13:20 of Monday, or anytime I'm in the lab.
Non-regionally-oriented course
Development of social and professional independence
- Course that cultivates an ability for utilizing knowledge
More than one class is interactive
Course by professor with work experience
| Work experience |
Work experience and relevance to the course content if applicable |
| N/A |
該当しない |
Education related SDGs:the Sustainable Development Goals
Last modified : Fri Mar 01 04:34:33 JST 2024