Algebra I

Academic unit or major

Mathematics

Instructor(s)

Masatoshi Suzuki

Class Format

Lecture (Face-to-face)

Media-enhanced courses

-

Day of week/Period
(Classrooms)

3-4 Wed / 3-4 Thu

Class

-

Course Code

ZUA.A301

Number of credits

200

Course offered

2024

Offered quarter

1-2Q

Syllabus updated

Mar 14, 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. The second half of the course will be devoted on basic topics surrounding Noetherian and Artinian rings, local rings, and homological algebras. This will be followed by a study on basic topics on tensor products of modules, right exactness of tensor products, and further related concepts (e.g., flatness).
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 mind 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.
4) Understand the notion of Noethrian and Artinian rings, and make use of fundamental operations for them correctly.
5) Understand the notion of local rings, and make use of fundamental operations for them correctly.
6) Understand homological algebra, and make use of fundamental operations for them correctly.

Keywords

rings, ideal, residue rings, modules, tensor products, localization, Noetherian rings, Artinian rings, local rings, homological algebra

Competencies

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

Class flow

Standard lecture course.

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.A301 ： Algebra I
• MTH.A302 ： Algebra II
• MTH.A201 ： Introduction to Algebra I
• MTH.A202 ： Introduction to Algebra II
• ZUA.A302 ： Exercises in Algebra B I

Prerequisites

Students must 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.
Students are strongly recommended to take ZUA.A302: Exercises in Algebra B I (if not passed yet) at the same time.

Other

None in particular.