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.
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.
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 |
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- | - | ✔ | - | - |
Standard lecture course accompanied by discussion session.
Course schedule | Required learning | |
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Class 1 | localization of rings and operations for ideals | Details will be provided during each class session. |
Class 2 | discussion session | Details will be provided during each class session. |
Class 3 | primary ideals and primary decomposition of ideals | Details will be provided during each class session. |
Class 4 | discussion session | Details will be provided during each class session. |
Class 5 | Noetherian rings and Hilbert's Basis Theorem | Details will be provided during each class session. |
Class 6 | discussion session | Details will be provided during each class session. |
Class 7 | modules over rings and free modules | Details will be provided during each class session. |
Class 8 | discussion session | Details will be provided during each class session. |
Class 9 | modules over a principal ideal domain and elementary divisor theory | Details will be provided during each class session. |
Class 10 | discussion session | Details will be provided during each class session. |
Class 11 | structure theorem for finitely generated modules | Details will be provided during each class session. |
Class 12 | discussion session | Details will be provided during each class session. |
Class 13 | Jordan canonical forms and how to derive them | 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
Ryoshi Hotta, Introduction to Algebra --Groups and Modules-- (in Japanese), Shoukabo
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, Introduction to Algebra I, II, III, IV, and Algebra I; or, they must have equivalent knowledge.
None in particular.