Algebra I

Academic unit or major

Undergraduate major in Mathematics

Instructor(s)

Masatoshi Suzuki

Class Format

Lecture/Exercise (Face-to-face)

Media-enhanced courses

-

Day of week/Period
(Classrooms)

3-6 Thu

Class

-

Course Code

MTH.A301

Number of credits

110

Course offered

2025

Offered quarter

1Q

Syllabus updated

Mar 19, 2025

Language

Japanese

Syllabus

Course overview and goals

The main topics of this course are the basic concepts and properties related to (commutative) rings, ideals, and modules over them. After reviewing some of the basics on (commutative) rings, ideals, and residue rings mod out by ideals, students will learn the concept of modules over a ring systematically, together with many of the related concepts including submodules, residue modules, linear mappings, homomorphism theorem, direct sums and direct products, exact sequences, Hom modules, free modules, etc. This will be followed by a study on basic topics on tensor products of modules, the right exactness of tensor products, and further related concepts (e.g., flatness). This course is followed by "Algebra II."

Rings, ideals, and modules over rings are among the most basic concepts in advanced algebra, which admits wide applications. However, its abstractness would cause several difficulties for newcomers. Students in this course will attempt to solidify these concepts in their minds by becoming familiar with these kinds of abstract concepts through rational integer rings and polynomial rings which are typical examples of (commutative) rings.

Course description and aims

By the end of this course, students will be able to:
1) Understand the notions of (commutative) rings and modules over rings.
2) Understand tensor products and make use of them correctly.
3) Understand localization and make use of them correctly.

Keywords

rings, ideals, residue rings, modules, tensor products, localization

Competencies

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

Class flow

Standard lecture course accompanied by discussion sessions.

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.

This will be announced during the lecture.

Evaluation methods and criteria

This will be announced during the lecture.

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.A302 ： Algebra II

Prerequisites

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

Other

None in particular.