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 basics on rings and modules.
  2. Understand 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 elementary number theory (GCD, LCM, Euclidean algorithm, etc.). 180minutes
3. Ideals and quotient rings Review elementary number theory (modular arithmetic). 180minutes
4. Fundamental theorem on homomorphisms Review the fundamental theorem on homomorphisms for groups (from Algebra I). 180minutes
5. Principal ideal domain (PID) Review the last lecture 180minutes
6. Unique factorization domain (UFD) Review the last lecture 180minutes
7. Mid-term exam Review the content of this course 270minutes
8. Rings and modules Review the notion of linear space 180minutes
9. Direct sums and projections Review the last lecture 180minutes
10. Elementary divisors (1) Introduction from linear algebra Review the last lecture 180minutes
11. Elementary divisors (2) Euclid domains Review the last lecture 180minutes
12. Fundamental theorem of finitely generated abelian groups Review the last lecture 180minutes
13. Jordan canonical forms Review the last lecture 180minutes
14. Final exam 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
Evaluate by the scores of the mid-term exam, the final exam and reports.
Textbooks and reference materials
Reference materials will be announced in the first lecture.
Prerequisites
Linear algebra I, II and Algebra I is assumed.
Office hours and How to contact professors for questions
  • 12:30-13:10 of Monday, or anytime I'm in the lab.
Relation to the environment
Non-environment-related course
Regionally-oriented
Non-regionally-oriented course
Development of social and professional independence
  • Non-social and professional independence development course
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 applicatable
N/A 該当しない
Last modified : Sat Oct 26 04:07:54 JST 2019