Algebra is a discipline of mathematics that deals with abstract notions which generalize algebraic operations on various mathematical objects. The main subjects of of this course include basic notions and properties of algebraic operations and of commutative rings, which are an abstraction/generalization of the integers and polynomials, and their ideals and residue rings. To help deeper understanding of the newly learnt concepts, each even-numbered class is devoted to a discussion session, where excercises are given related to the contents of the preceding lecture. This course succeeds ``Introduction to Algebra I'' offered in the first quater.
The contents of this course form not only a foundation of the whole Algebra but also an indispensable body of knowledge in other areas of mathematics such as Analysis and Geometry. Also, it is a basic attitude in all mathematical sciences to perform logical arguments without depending on intuition. In this course, we provide rigorous proofs, based on the notions of sets and maps, so that the students can learn how typical mathematical arguments should go.
To become familiar with important notions such as the axiom of rings, subrings, ideals, residue rings, homomorphisms of rings, and the fundamental theorem on ring homomorphisms.
To become able to prove by him/herself basic properties of these objects.
ring, subring, ideal, residue ring, homomorphism of rings, the fundamental theorem on ring homomorphims
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 sesssions.
Course schedule | Required learning | |
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Class 1 | Axiom of rings, tyical examples of rings, and first properties of rings | Details will be provided during each class session. |
Class 2 | Discussion session on the axiom of rings, tyical examples of rings, and first properties of rings | Details will be provided during each class session. |
Class 3 | Basic properties of the zero and inverse elements of a ring | Details will be provided during each class session. |
Class 4 | Discussion session on basic properties of the zero and inverse elements of a ring | Details will be provided during each class session. |
Class 5 | Definition of a subring, criterion for subrings, and examples of subrings | Details will be provided during each class session. |
Class 6 | Discussion session on the definition of a subring, criterion for subrings, and examples of subrings | Details will be provided during each class session. |
Class 7 | Homomorphisms of rings and their basic properties | Details will be provided during each class session. |
Class 8 | Discussion session on homomorphisms of rings and their basic properties | Details will be provided during each class session. |
Class 9 | Ideals of a ring | Details will be provided during each class session. |
Class 10 | Discussion session on ideals of a ring | Details will be provided during each class session. |
Class 11 | Residue rings and the first fundamental theorem on ring homomorphisms | Details will be provided during each class session. |
Class 12 | Discussion session on residue rings and the first fundamental theorem on ring homomorphisms | Details will be provided during each class session. |
Class 13 | The second and third fundamental theorems on ring homomorphisms | Details will be provided during each class session. |
Class 14 | Discussion session on the second and third fundamental theorems on ring homomorphisms | Details will be provided during each class session. |
Class 15 | Checking session | Details will be provided during each class session. |
Shoichi Nakajima : Basics of Algebra and Arithmetic, Kyoritsu Shuppan, Co., Ltd., 2000.
P.J. Cameron : Introduction to Algebra (second ed.), Oxford Univ. Press, 2008.
N. Jacobson : Basic Algebra I (second ed.), Dover，1985.
M. Artin : Algebra (second ed.), Addison-Wesley, 2011.
N. Herstein: Topics in algebra, John Wiley & Sons, 1975.
A. Weil: Number Theory for Beginners, Springer-Verlag, 1979.
Based on evaluation of the results for discussion session and final examination. Details will be announced during a lecture.
Students are supposed to have completed [Linear Algebra I / Recitation], [Linear Algebra II], [Linear Algebra Recitation II] and [Introduction to Algebra I].