Algebra II

Academic unit or major

Undergraduate major in Mathematics

Instructor(s)

Fumiharu Kato / Mutsuro Somekawa

Class Format

Lecture/Exercise

Media-enhanced courses

-

Day of week/Period
(Classrooms)

3-4 Wed (H103) / 5-6 Thu (H103)

Class

-

Course Code

MTH.A302

Number of credits

110

Course offered

2021

Offered quarter

2Q

Syllabus updated

Jul 10, 2025

Language

Japanese

Syllabus

Course overview and goals

The main topics of this course are more advanced concepts and properties of (commutative) rings and their ideals, as well as the concepts and properties of modules over (Noetherian) rings. The instructor in this course will first explain localization of (commutative) rings, and several fundamental operations for ideals, then explain primary decomposition of ideals. Next, the instructor will introduce the concept of Noetherian rings, explain several of their properties, and furthermore Hilbert's basis theorem. Finally, the instructor will introduce the concept of modules over (Noetherian) rings, explaining in particular the structure theorem of finitely generated modules over principal ideal domains. Then students will learn as an application about the existence of Jordan normal forms that are very useful in linear algebra, as well as how to calculate them. In each class, students will complete exercises related to the course content. This course follows ""Algebra 1"".
The theory of modules over rings expands and develops the theory of vector spaces and linear mapping theory learned in linear algebra to more general cases. In addition, the concept of (Noetherian) rings and modules over them is a fundamental concept of algebra, and a very wide range of applications extending over both algebra and mathematics as a whole. The goal of this course is for students to become familiar with these concepts, firmly grasp their basic properties, and learn to use them correctly.

Course description and aims

By the end of this course, students will be able to:
1) Understand the notion of localization of (commutative) rings, and make use of fundamental operations for ideals correctly.
2) Understand the primary decomposition of ideals, and make use of it.
3) Explain the definition and some of the basic properties of a Noetherian ring.
4) Understand the notion of modules over a (Noetherian) ring and some of their properties.
5) Understand and make use of the structure theorem for finitely generated modules over a principal ideal domain correctly.

Keywords

localization of rings, primary ideal, Noetherian ring, Hilbert's Basis Theorem, module over a ring, module over a principal ideal domain, elementary divisor, finitely generated module, Jordan canonical form

Competencies

• Specialist skills
• Intercultural skills
• Communication skills
• Critical thinking skills
• Practical and/or problem-solving skills

Class flow

Standard lecture course accompanied by discussion session.

Course schedule/Objectives

Study advice (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

Evaluation methods and criteria

TBA

Related courses

• MTH.A201 ： Introduction to Algebra I
• MTH.A202 ： Introduction to Algebra II
• MTH.A203 ： Introduction to Algebra III
• MTH.A204 ： Introduction to Algebra IV
• MTH.A301 ： Algebra I

Prerequisites

Students are required to have successfully completed Linear Algebra I/Recitation, Linear Algebra II, Linear Algebra Recitation II, Advanced Linear Algebra I, II, Introduction to Algebra I, II, III, IV, and Algebra I; or, they must have equivalent knowledge.

Other

None in particular.