Course title
330243002
Algebra 2

 SHIMIZU Kenichi
Middle-level Diploma Policy (mDP)
Program / Major mDP Goals
Mathematical Sciences Course DP-4・2 キャリアを見据えた高度な専門知識
現象の背後にある数理構造やデータのパターンを理論的に解析し、社会や自然科学、工学における複雑な課題に対して数理的視点から解決戦略を提案できる。
Purpose of class
Understand basics on rings and modules.
Course description
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”.
Goals and objectives
  1. Understand and be able to explain basics on rings and modules.
  2. Understand and be able to explain basics on the division relation in a ring.
  3. 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. 5% 25% 0% 30%
2. 5% 15% 15% 35%
3. 10% 25% 35%
Total. 20% 40% 40% -
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.
Language
Japanese
Class schedule

 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
Feedback on exams, assignments, etc.
ways of feedback specific contents about "Other"
Feedback in the class
Textbooks and reference materials
None
Prerequisites
Office hours and How to contact professors for questions
  • 12:30-13:20 of Monday, or anytime I’m in the lab.
Regionally-oriented
Non-regionally-oriented course
Development of social and professional independence
  • Course that cultivates an ability for utilizing knowledge
Active-learning course
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 : Mon Mar 23 04:06:59 JST 2026