The main subjects of this course are basic and some advanced notions in (commutative) ring theory, and also the notion of modules over a (Noetherian) ring.
In this course, we first introduce various kinds of ideals such as principal ideals, prime ideals, and maximal ideals, and then explain the definition of quotient rings by these ideals; also, we explain some of their basic properties. Next, we introduce the notion of homomorphisms between rings, and then explain First Isomorphism Theorem and Chinese remainder theorem. Next, we introduce the notion of Euclidean domain and (more generally) the notion of principal ideal domain, and then explain some of the properties of prime elements and irreducible elements of a unique factorization domain; also, we explain that every principal ideal domain is a unique factorization domain. Next, we introduce the notion of localization of (commutative) rings, and then explain some fundamental operations for ideals; also, we explain the primary decomposition of an ideal. Next, we introduce the notion of Noetherian rings, and then explain some of their properties; in particular, we explain Hilbert's basis theorem. Finally, we introduce the notion of modules over a (Noetherian) ring, and then explain the structure theorem for finitely generated modules over a principal ideal domain. As an application, we explain how to derive the Jordan normal form of a given matrix, which is very useful in linear algebra.
The notions of ring, ideal, and quotient ring are most fundamental in algebra, and are applicable to describe a wide variety of objects. However, it is not easy for beginners to comprehend these abstract notions without suitable training. In this course, we often take up the ring of rational integers and the ring of polynomials as examples, and explain the abstract notions above by using these concrete examples, so that students get familiar with them. Now, the theory of modules over a ring can be thought of as a generalization and a further development of linear algebra, which is the theory of vector spaces and linear mappings. Also, the notions of Noetherian ring and modules over it are most fundamental in algebra,and are applicable to describe a wide variety of objects not only in algebra, but also in the whole mathematics. Another aim of this course is to make students familiar with these notions, understand some of their basic properties, and enable to make use of them correctly.
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.
5) Understand the notion of localization of (commutative) rings, and make use of fundamental operations for ideals correctly.
6) Understand the primary decomposition of an ideal, and make use of it.
7) Explain the definition and some of the basic properties of a Noetherian ring.
8) Understand the notion of modules over a (Noetherian) ring and some of their properties.
9) Understand and make use of the structure theorem for finitely generated modules over a principal ideal domain correctly.
ring, ideal, principal ideal, quotient ring, prime ideal, maximal ideal, principal ideal domain, unique factorization domain,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
Intercultural skills | Communication skills | ✔ Specialist skills | Critical thinking skills | Practical and/or problem-solving skills |
Standard lecture course.
Course schedule | Required learning | |
---|---|---|
Class 1 | units, zero-divisors, nilpotent elements, and integral domains | Details will be provided during each class session. |
Class 2 | ideals and principal ideals | Details will be provided during each class session. |
Class 3 | prime ideals, maximal ideals, and quotient rings | Details will be provided during each class session. |
Class 4 | First Isomorphism Theorem and Chinese remainder theorem | Details will be provided during each class session. |
Class 5 | Euclidean domains | Details will be provided during each class session. |
Class 6 | principal ideal domains | Details will be provided during each class session. |
Class 7 | unique factorization domains, prime elements, and irreducible elements | Details will be provided during each class session. |
Class 8 | Evaluation of progress | Details will be provided during each class session. |
Class 9 | localization of rings and operations for ideals | Details will be provided during each class session. |
Class 10 | primary ideals and primary decomposition of an ideal | Details will be provided during each class session. |
Class 11 | Noetherian rings and Hilbert's basis theorem | Details will be provided during each class session. |
Class 12 | modules over a ring and free modules | Details will be provided during each class session. |
Class 13 | modules over a principal ideal domain and elementary divisor theory | Details will be provided during each class session. |
Class 14 | structure theorem for finitely generated modules | Details will be provided during each class session. |
Class 15 | Jordan canonical forms and how to derive them | Details will be provided during each class session. |
None
Shigemoto Asano, Algebra I: Basic Notions, Rings, and Modules (in Japanese), Morikita shuppan
Shouichi Nakajima, Elements of Algebra and Arithmetic (in Japanese), Kyoritsu shuppan
Ryoshi Hotta, Introduction to Algebra --Groups and Modules-- (in Japanese), Shoukabo
Midterm exam. (about 50 %), final exam. (about 50 %).
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.
None in particular.