### 2016　Algebra I

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Instructor(s)
Naito Satoshi
Course component(s)
Lecture / Exercise
Day/Period(Room No.)
Thr3-6(H103)
Group
-
Course number
MTH.A301
Credits
2
2016
Offered quarter
1Q
Syllabus updated
2017/1/11
Lecture notes updated
-
Language used
Japanese
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### Course description and aims

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.

### Student learning outcomes

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.

### Keywords

ring, ideal, principal ideal, quotient ring, prime ideal, maximal ideal, principal ideal domain

### Competencies that will be developed

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

### Class flow

Standard lecture course accompanied by discussion session.

### Course schedule/Required learning

Course schedule Required learning
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.

### Textbook(s)

Shoichi Nakajima, Elements of Algebra and Arithmetic, Kyoritsu (Japanese)

None

### Assessment criteria and methods

Final exam. 70%, discussion session 30%.

### Related courses

• MTH.A201 ： Introduction to Algebra I
• MTH.A202 ： Introduction to Algebra II
• MTH.A203 ： Introduction to Algebra III
• MTH.A204 ： Introduction to Algebra IV
• MTH.A302 ： Algebra II

### Prerequisites (i.e., required knowledge, skills, courses, etc.)

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.