# COURSE OUTLINE: MH4200

Course Title

Course Code

### MH4200

Offered Study Year 4, Semester 1
Course Coordinators Wu Guohua (Assoc Prof) guohua@ntu.edu.sg 6513 7186
Wang Huaxiong (Assoc Prof) hxwang@ntu.edu.sg 6513 7472
Pre-requisites MH1201 and MH3200
AU 4
Contact hours Lectures: 39, Tutorials: 12
Approved for delivery from AY 2021/22 semester 1
Last revised 6 Sep 2021, 11:24

### Course Aims

This mathematics course, which is a prescribed elective for students in the pure mathematics track, aims at understanding basics in communicative algebra, which captures concepts in ring theory and module theory. This course will lead you to various structures that are common to diverse mathematical objects, and learn how to apply the abstract theory to different mathematical contexts. It will be essential for advanced courses in algebraic number theory, algebraic geometry, module theory, and anyone who will need to work with number theory, geometry, coding theory and cryptography.

### Intended Learning Outcomes

Upon successfully completing this course, you should be able to:

1. State basic definitions related to rings, modules and short exact sequences, and explain them to a layman
2. Explain the Hilbert's basis theorem, and Hilbert's nullstellensatz, and their applications in polynomial rings and field theory
3. Compute Grobner basis via Buchberger's algorithm
4. Classify finitely generated abelian groups by applying the fundamental theorem
5. Prove or disprove statements related to commutative rings and modules

### Course Content

Sums, products and radicals of ideals of communicative rings

Noetherian rings and Artinian rings, Hilbert's basis theorem

Prime ideals, maximal ideals, Localization of rings and Hilbert's Nullstellensatz

Grobner basis and Buchberger's algorithm

Modules, submodules, quotient modules

Isomorphism theorems for modules

Direct sum of modules, and fundamental theorem for finitely generated abelian groups

Projective modules, injective modules and short exact sequences

### Assessment

Component Course ILOs tested SPMS-MAS Graduate Attributes tested Weighting Team / Individual Assessment Rubrics
Continuous Assessment
Tutorials
Presentation 3, 4, 5 1. b
3. a
4. a
5. a
15 individual See Appendix for rubric
Test 1, 2, 3 1. b
5. a
15 individual See Appendix for rubric
Mid-semester Quiz
Short Answer Questions 1, 2, 3 1. b
3. a
20 individual See Appendix for rubric
Examination (2 hours)
Short Answer Questions 1, 2, 3, 4, 5 1. b
3. a
50 individual See Appendix for rubric
Total 100%

These are the relevant SPMS-MAS Graduate Attributes.

### 1. Competence

b. Formulate mathematical statements precisely using rigorous mathematical language

### 3. Communication

a. Present mathematics ideas logically and coherently at the appropriate level for the intended audience

### 4. Civic-mindedness

a. Develop and communicate mathematical ideas and concepts relevant in everyday life for the benefits of society

### 5. Character

a. Act in socially responsible and ethical ways in line with the societal expectations of a mathematics professional, particularly in relation to analysis of data, computer security, numerical computations and algorithms

### Formative Feedback

Attending lectures, reading lecture notes and related chapters in reference book, and working through weekly tutorial problems and discussing with instructors and peers, you will be able meet learning outcomes 1,2,3,4,5.

After each test, instructors will discuss the test solution to highlight strength and weakness of students. The Examiner's report will be uploaded to NTULearn to provide feedback on the strength and weakness of students in the module.

### Reading and References

Reference: Abstract Algebra, Third Edition, by David S. Dummit and Richard M. Foote, John Wiley and Sons, Inc.

ISBN 10: 0470386282 / ISBN 13: 9780470386286

### Course Policies and Student Responsibilities

You are expected to attend all classes punctually and take all scheduled assignments, tests, presentations by due dates. You are expected to take
responsibility to follow up with course notes, assignments and course related
announcements. You are expected to participate in all discussions, presentations and activities. You are expected to attend the midterms. In case of medical leaves, You should provide a satisfying medical certificate on time.

Good academic work depends on honesty and ethical behaviour. The quality of your work as a student relies on adhering to the principles of academic integrity and to the NTU Honour Code, a set of values shared by the whole university community. Truth, Trust and Justice are at the core of NTU’s shared values.

### Course Instructors

Instructor Office Location Phone Email
Wang Huaxiong (Assoc Prof) SPMS-MAS-05-26 6513 7472 hxwang@ntu.edu.sg
Wu Guohua (Assoc Prof) SPMS-MAS-05-40 6513 7186 guohua@ntu.edu.sg

### Planned Weekly Schedule

Week Topic Course ILO Readings/ Activities
1

Sums, products and radicals of ideals of communicative rings

1, 2, 5

Chapter 15, sections 1-3

2

Prime ideals, maximal ideals, Localization of rings , Nilradical and Jacobson radical

1, 2, 5

Chapter 15, section 4-5

3

Introduction to Module Theory (1)

1, 4, 5

Chapter 10, sections 1-3

4

Introduction to Module Theory (2)

1, 4, 5

Chapter 10, sections 1-3

5

Noetherian rings and Artinian rings

1, 2, 5

Chapter 15, section 1
Chapter 16, section 1

6

Noetherian modules and Artinian modules, Hilbert's Basis Theorem

1, 2, 5

Chapter 9, section 6

7

Grobner Basis and Buchberger's algorithm

1, 2, 3, 5

Chapter 9, section 6

8

Integral extensions, Hilbert's Nullstellensatz, Noetherian Normalization Theorem

1, 2, 5

Chapter 15, section 3

9

Projective modules, injective modules and short exact sequences (1)

1, 4, 5

Chapter 10, sections 4-5

10

Projective modules, injective modules and short exact sequences (2)

1, 4, 5

Chapter 10, sections 4-5

11

Direct sum of modules

1, 4, 5

Chapter 10, sections 4-5

12

Fundamental theorem for finitely generated abelian groups (1)

1, 4, 5

Chapter 5, sections 1-3

13

Fundamental theorem for finitely generated abelian groups (2)

1, 4, 5

Chapter 5, sections 1-3

### Appendix 1: Assessment Rubrics

#### Rubric for Tutorials: Presentation (15%)

 Criteria Fail to meet  standard (0 - 3) (Prestructural) Nearly Meet Standard (4 – 6) (Unistructural) Meet  Standard  (7 - 9) (Multi-structural) Exceed Standard  (10 - 12) (Relational) Far Exceed Standard (13 - 15) (Extended) Organization and Structure No clear structure showing in the presentation. Topics presented are incoherent. No clear distinction between parts on motivation, main results, and conclusion. Somewhat structured presentation Distinct parts on motivation, main results, conclusions, etc. are indicated but the contents in each part are unstructured Structured presentation Distinct parts on motivation, results, conclusions, etc. are indicated, with contents mostly organized. Key conclusions are not specified in a direct way Well-structured presentation Distinct parts on motivation, results, conclusions, etc. are indicated and the contents in each part content is well- structured The key conclusions are clear Above standard structured presentation. Content of motivation, results, conclusions, etc. are well organized throughout the entire presentation. The presented contents are not only comprehensive, and the presentation is efficient. Analytical Thinking Part of the understanding of concepts, methods and proofs are not correct The understanding are correct, but could not provide the motivation behind these concepts and basic ideas of proofs The understanding are correct and clear, and can provide motivation behind these concepts and proofs. Need to show the relations between these and other relevant topics The understanding are correct and clear, and can provide enough knowledge of the role of these topics in the whole subject. The presentation does not mention the current research on these topics The understanding are correct and clear, and can provide thorough knowledge of the position of these topics in the whole subject. The presentation digs into the updated research on the topics presented Effectiveness of oral presentation and Q&A Does not present ideas effectively, (pace, tone and presentation style) and cannot keep the attention of audience through most of the presentation. Communicates ideas somewhat effectively (pace, tone and style). Maintain the attention of audience in some parts of the presentation. Communicate ideas in an effective and understandable manner throughout most of the presentation. Motivate the interest of the audience and keep their attention for the major part of the presentation Explain difficult methods in an effective and understandable way. Motivate interest of the audience, and engage the audience throughout the presentation Far exceed expectations of a fourth year student in communicating complex scientific concepts. Motivate the interest to further topics extending the presented contents Individual Contribution and Q&A No effective contribution in the presentation The Q&A session shows little knowledge on the chosen topic. Some contribution of proofs and methods shown in the presentation The Q&A session only shows one aspect of the chosen topic and shows limited knowledge of the chosen topic Interesting contribution of proofs and methods shown in the presentation The Q&A session reflects more than one aspect of the chosen topic, but does not show connection with related topics. Provide motivation of proofs and methods shown in the presentation The Q&A session reflects depth of knowledge, not only in an individual topic, but in the whole area. Provide new proofs and methods shown in the presentation The Q&A session reflects coherence, fluency, and depth of knowledge in the whole area. Show new perspectives on the topic presented.

#### Rubric for Tutorials: Test (15%)

Point-based Marking. Marks will be awarded to analytical, logical and correct justification, or

computation, or examples

#### Rubric for Mid-semester Quiz: Short Answer Questions (20%)

Point-based Marking. Marks will be awarded to analytical, logical and correct justification, or

computation, or examples

#### Rubric for Examination: Short Answer Questions (50%)

Point-based Marking. Marks will be awarded to analytical, logical and correct justification, or

computation, or examples