Course Title | ## Algebra II | ||

Course Code | ## MH3220 | ||

Offered | Study Year 3, Semester 1 | ||

Course Coordinators | Ng Keng Meng (Assoc Prof) | kmng@ntu.edu.sg | 6513 8656 |

Lim Kay Jin (Dr) | limkj@ntu.edu.sg | 6513 7462 | |

Pre-requisites | MH1201 and MH2220 | ||

Mutually exclusive | MH3200 | ||

AU | 4 | ||

Contact hours | Lectures: 39, Tutorials: 12 | ||

Approved for delivery from | AY 2023/24 semester 1 | ||

Last revised | 13 Oct 2022, 09:47 |

This MAS course aims to introduce ring theory that is essential for more advanced algebra courses and applications. The axiomatic concepts serve as a language to study concrete examples in broader sense and helps in developing logical thinking.

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

- State basic definitions related to rings and explain them to a layman.
- Give examples and counter-examples involving rings.
- Categorize rings based on their properties.
- Compute quotient rings.
- Distinguish ideals from subrings.
- Apply ring isomorphism theorems to prove that two rings are isomorphic.
- Prove or disprove statements involving rings.

Basic definitions from ring theory

Examples of rings including polynomial rings, matrix rings

Ideals and ring homomorphisms

Maximal and prime ideals

Quotient rings and isomorphism theorems

Ring of fractions

Euclidean domains, Principal ideal domains, Unique factorization domains

Component | Course ILOs tested | SPMS-MAS Graduate Attributes tested | Weighting | Team / Individual | Assessment Rubrics |
---|---|---|---|---|---|

Continuous Assessment | |||||

Mid-semester Quiz | |||||

Short Answer Questions | 1, 2, 3, 4, 7 | 1. a, b, c2. b | 25 | individual | See Appendix for rubric |

Short Answer Questions 1 | 2, 3, 4, 5, 6, 7 | 1. a, b, c2. b | 25 | individual | See Appendix for rubric |

Examination (2 hours) | |||||

Short Answer Questions | 1, 2, 3, 4, 5, 6, 7 | 1. a, b, c2. b | 50 | individual | See Appendix for rubric |

Total | 100% |

These are the relevant SPMS-MAS Graduate Attributes.

## 1. Competence

a. Independently process and interpret mathematical theories and methodologies, and apply them to solve problems

b. Formulate mathematical statements precisely using rigorous mathematical language

c. Discover patterns by abstraction from examples

## 2. Creativity

b. Build on the connection between subfields of mathematics to tackle new problems

You will receive formative feedback for your CA. It is done by commenting on the mistakes and misunderstanding that appeared in your CA. This is possible because this course has a small number of students (typically less than 20). General feedback for your performance as part of the end of course review is done based on the final exam: the most common mistakes, as well as the questions that were best answered, are discussed in the report provided to all students.

Lectures (39 hours) | Interactive Lecture: |

Tutorials (12 hours) | Problem solving: |

D.S. Dummit, R.M. Foote, Abstract Algebra, third edition, John Wiley & Sons, Inc., Hoboken, NJ, 2004.

ISBN-10 : 0471433349

You are expected to attend all classes punctually and take all scheduled assignments and tests 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 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.

As a student, it is important that you recognize your responsibilities in understanding and applying the principles of academic integrity in all the work you do at NTU. Not knowing what is involved in maintaining academic integrity does not excuse academic dishonesty. You need to actively equip yourself with strategies to avoid all forms of academic dishonesty, including plagiarism, academic fraud, collusion and cheating. If you are uncertain of the definitions of any of these terms, you should go to the Academic Integrity website for more information. Consult your instructor(s) if you need any clarification about the requirements of academic integrity in the course.

Instructor | Office Location | Phone | |
---|---|---|---|

Ng Keng Meng (Assoc Prof) | MAS-05-09 | 6513 8656 | kmng@ntu.edu.sg |

Lim Kay Jin (Dr) | SPMS-MAS-05-16 | 6513 7462 | limkj@ntu.edu.sg |

Week | Topic | Course ILO | Readings/ Activities |
---|---|---|---|

1 | Basic definitions from ring theory | 1 | Textbook: 7.1 |

2 | Examples of rings including polynomial rings, matrix rings | 1, 2 | Textbook: 7.2 |

3 | Ideals | 1, 2, 3, 5, 7 | Textbook: 7.4 |

4 | Properties of ideals | 1, 2, 3, 5, 7 | Textbook: 7.4 |

5 | Maximal and prime ideals | 1, 2, 3, 5, 7 | Textbook: 7.4 |

6 | Ring homomorphisms | 3, 7 | Textbook: 7.3 |

7 | Quotient rings | 3, 4, 7 | Textbook: 7.3 |

8 | Isomorphism theorems | 1, 2, 3, 4, 5, 6, 7 | Textbook: 7.3 |

9 | Isomorphism theorems | 1, 2, 3, 4, 5, 6, 7 | Textbook: 7.3 |

10 | Ring of fractions | 1, 2 | Textbook: 7.5 |

11 | Polynomial Ring | 1, 2 | Textbook: 9.1, 9.2 |

12 | Euclidean domains, Principal ideal domains, Unique factorization domains | 1, 2, 3, 5, 6, 7 | Textbook: 8.1, 8.2, 8.3 |

13 | Euclidean domains, Principal ideal domains, Unique factorization domains | 1, 2, 3, 5, 6, 7 | Textbook: 8.1, 8.2, 8.3 |