The main topics of this course are (commutative) rings and the basic concepts and properties related to their ideals. Students in this course will learn first about (commutative) rings and various ideals (principal ideals, prime ideals, maximal ideals, etc.), and the basic properties of quotient rings resulting from them. Next, students will learn about the concept of homomorphism between rings, the homomorphism theorem, and the Chinese remainder theorem. Finally, students will learn about several properties of Euclidean domains and (generalized) principal ideal domains, as well as concepts of prime elements and irreducible elements in unique factorization domains. This course is followed by "Algebra II".
Rings, their ideals, and quotient rings are the most fundamental concepts in algebra, with a very wide range of applications. On the other hand, they are abstract concepts which many beginners have a difficult time understanding. 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 (commutative) rings.
By the end of this course, students will be able to:
1) Understand the notions of ideal, principal ideal, prime ideal, maximal ideal, and quotient ring.
2) Understand First Isomorphism Theorem and Chinese remainder theorem, and make use of them correctly.
3) Explain some of the basic properties of an Euclidean domain and (more generally) of a principal ideal domain.
4) Understand the notions of prime elements and irreducible elements of a unique factorization domain, and use them correctly.
ring, ideal, principal ideal, quotient ring, prime ideal, maximal ideal, principal ideal domain
Intercultural skills | Communication skills | Specialist skills | Critical thinking skills | Practical and/or problem-solving skills |
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- | - | ✔ | - | - |
Standard lecture course accompanied by discussion session.
Course schedule | Required learning | |
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Class 1 | units, zero-divisors, nilpotent elements, integral domains | Details will be provided during each class session. |
Class 2 | discussion session | Details will be provided during each class session. |
Class 3 | ideals and principal ideals | Details will be provided during each class session. |
Class 4 | discussion session | Details will be provided during each class session. |
Class 5 | prime ideals, maximal ideals, and quotient rings | Details will be provided during each class session. |
Class 6 | discussion session | Details will be provided during each class session. |
Class 7 | First Isomorphism Theorem and Chinese Remainder Theorem | Details will be provided during each class session. |
Class 8 | discussion session | |
Class 9 | Euclidean domains | Details will be provided during each class session. |
Class 10 | discussion session | Details will be provided during each class session. |
Class 11 | principal ideal domain | Details will be provided during each class session. |
Class 12 | discussion session | Details will be provided during each class session. |
Class 13 | unique factorization domain, prime elements and irreducible elements | Details will be provided during each class session. |
Class 14 | discussion session | Details will be provided during each class session. |
Class 15 | evaluation of progress | Details will be provided during each class session. |
Shigemoto Asano, Algebra I: Basic Notions, Rings, and Modules (in Japanese), Morikita shuppan
Shoichi Nakajima, Elements of Algebra and Arithmetic (in Japanese), Kyoritsu shuppan
Final exam. 70%, discussion session 30%.
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.
None in particular.