### 2021　Exercises in Algebra B I

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Academic unit or major
Mathematics
Instructor(s)
Kato Fumiharu  Kawachi Takeshi  Somekawa Mutsuro
Course component(s)
Exercise    (ZOOM)
Day/Period(Room No.)
Thr5-6(H103)
Group
-
Course number
ZUA.A302
Credits
2
Academic year
2021
Offered quarter
1-2Q
Syllabus updated
2021/3/19
Lecture notes updated
-
Language used
Japanese
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### Course description and aims

This course is an exercise session for the lecture course "Algebra I (ZUA.A301)". The materials for exercise are chosen from that course.

### 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.
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.

### Keywords

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

### Competencies that will be developed

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

### Class flow

Students are given exercise problems related to what is taught in the course "Algebra I".

### Course schedule/Required learning

Course schedule Required learning
Class 1 Discussion session on the following materials: units, zero-divisors, nilpotent elements, and integral domains. Details will be provided during each class session.
Class 2 Discussion session on the following materials: ideals and principal ideals. Details will be provided during each class session.
Class 3 Discussion session on the following materials: prime ideals, maximal ideals, and quotient rings. Details will be provided during each class session.
Class 4 Discussion session on the following materials: First Isomorphism Theorem and Chinese remainder theorem. Details will be provided during each class session.
Class 5 Discussion session on the following materials: Euclidean domains. Details will be provided during each class session.
Class 6 Discussion session on the following materials: principal ideal domains. Details will be provided during each class session.
Class 7 Discussion session on the following materials: unique factorization domains, prime elements, and irreducible elements. Details will be provided during each class session.
Class 8 Discussion session on the following materials: localization of rings and operations for ideals. Details will be provided during each class session.
Class 9 Discussion session on the following materials: primary ideals and primary decomposition of an ideal. Details will be provided during each class session.
Class 10 Discussion session on the following materials: Noetherian rings and Hilbert's basis theorem. Details will be provided during each class session.
Class 11 Discussion session on the following materials: modules over a ring and free modules. Details will be provided during each class session.
Class 12 Discussion session on the following materials: modules over a principal ideal domain and elementary divisor theory. Details will be provided during each class session.
Class 13 Discussion session on the following materials: structure theorem for finitely generated modules. Details will be provided during each class session.
Class 14 Discussion session on the following materials: Jordan canonical forms and how to derive them. Details will be provided during each class session.

### Out-of-Class Study Time (Preparation and Review)

To enhance effective learning, students are encouraged to spend approximately 100 minutes preparing for class and another 100 minutes reviewing class content afterwards (including assignments) for each class.
They should do so by referring to textbooks and other course material.

TBA

TBA

TBA

### Related courses

• MTH.A301 ： Algebra I
• MTH.A302 ： Algebra II
• MTH.A201 ： Introduction to Algebra I
• MTH.A202 ： Introduction to Algebra II
• ZUA.A301 ： Algebra I

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

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.A301: Algebra I (if not passed yet) at the same time.

### Other

None in particular.