Course title
V01354002
Algebra 2

 SHIMIZU Kenichi
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".
Purpose of class
Understand basics on rings and modules.
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.
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 (1) Review the last lecture 180minutes
4. Ideals and quotient rings (2) Review the last lecture 180minutes
5. Principal ideal domain (PID) Review the last lecture 180minutes
6. Unique factorization domain (UFD) Review the last lecture 180minutes
7. Principal ideal domains (PID) 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 (1) Review the last lecture 180minutes
14. Finitely generated modules over a PID (2) Review the content of this course 270minutes
Total. - - 2700minutes
Relationship between 'Goals and Objectives' and 'Course Outcomes'

 Mid-term exam Final exam Reports Total.
1. 20% 10% 5% 35%
2. 20% 10% 5% 35%
3. 20% 10% 30%
Total. 40% 40% 20% -
Evaluation method and criteria
Evaluated by the total score of homework.
Criteria for passing: Understand basics on rings and modules.
Textbooks and reference materials
None
Prerequisites
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.
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 : Sat Mar 19 00:04:25 JST 2022