# MA3201 Rings and modules

# News

14.08: First lecture Tuesday 22 of August, at 14:15 in K26 (are located in Kjemiblokk 4, 1.etg).

# General information

**Lecturer:** Sverre SmalĂ¸, email: sverresm at math ntnu no, SBII, room 850, tlf. 735 91750

**Office hours:** Tuesday 13:15. I am awailable all days between 9.00 to 15.00 from August 22nd on and until December ?th. (Only by appointment for Saturdays and Sundays)

Teaching hours:

Lectures: | Tuesday 14:15-16:00, K26 |

Friday 14:15-16:00, R60 |

Some of the Friday-lectures will be used as exercise sessions.

# Textbook and syllabus

Textbook

Authors: | P. B. Bhattacharya, S. K. Jain and S. R. Nagpaul |

Title: | Basic Abstract Algebra |

Publisher: | Cambridge University Press |

Edition: | Second Edition |

ISBN: | 0-521-46629-6 |

Preliminary syllabus:

Chapter 9 | All sections |

Chapter 10 | All sections |

Chapter 14 | 14.1-14.5 |

Chapter 19 | 19.1-19.3 |

Chapter 20 | All sections |

Chapter 21 | All sections |

# Plan/log

Date | Themes | Reference | |
---|---|---|---|

22.08 | Introduction, definitions, various examples including ring of polynomials, rings of formal power series, rational functions and Laurent series. | Chap 9.1, 9.2, 9.3 | |

25.08 | Subrings, characteristic of a ring, idempotent og nilpotent elements, ideals and homomorphisms | Chap 9.4, 9.5, 10.1, 10.2 | |

29.08 | More on Ideals and homomorphisms, left ideals, right ideals, ideals in the full matrix algebra, Intersections and sums af ideals, | Chap 10.1, 10.2, 10.3 | |

01.09 | Exercise 1 in the first hour! Direct sum of Ideals, Minimal, maximal and prime ideals, the real numbers as the rings of Cauchy-sequences modulo the ideal of sequences converging to zero | Chap 10.3, 10.4 | |

05.09 | The Chinese remainder theorem, Maximal ideals are prime, the maximal ideals of Z, C[x] and R[X], description of prime ideals for commutative rings. | Chap 10.5,10.6 | |

08.09 | Nilpotent and nil ideals, Zorns's lemma and the existence of maximal ideals. Modules and Vectorspaces | Chap 14.1, 14.2 | |

12.09 | The rest of Chapter 14.2, and most of 14.3 | Chap 14.2, 14.3 | |

15.09 | Exercises 2 and 3, the rest of 14.3 and start of 14.4 | Chap 14.3, 14.4 | |

19.09 | Rest of 14.4 and started on 14.5 | Chap 14.4, 14.5 | |

22.09 | Finished 14.4 and 19.1, Started on 19.2 | Chap 14.5, 19.1 19.2 | |

26.09 | Chap 19.2 | ||

29.09 | We will take a look at the exercises, so look through the problems in exercise 4. Any question from exercise 1, 2 and 3 can also be raised. | ||

03.10 | Endomorphism ring of a direct sum as a matrix ring, Th 19.2.7, 19.2.8,19.2.9, 19.2.10 | Chap 19.1, 19.2 | |

06.10 | Modules of finte length and composition series.th 19.2.11 | ||

10.10 | Wedderburn-Artin + examples | Chap 19.3 | |

13.10 | Exercise 5 and 6 | ||

17.10 | More examples illustarting Wedderburn-Artin's theorem and modules of finite length. Prove that for semisimple modules artinian and noetherian is the same | ||

20.10 | Exercise 6 | ||

24.10 | Smith normal form | Chapter 20.1,2,3 | |

27.10 | Exercise 6, 7, 8 | ||

31.10 | Applications of Smith normal form | Chapter 21.1,2,3 | |

03.11 | Exercise 8, 9 | ||

07.11 | Finishing Rational canonical forms and Jordan canonical forms | Chapter 21.4,5 | |

10.11 | Do the exam from Dec. 2016 | ||

14.11 | Summary |

# Exam

Final exam: 18.12.2016, written, 4 hours, 9:00-13:00

# Old Exams

# Problem sessions(tentative)

Exercise 1 | Chapter 9, page 173-176: | 1, 2, 3, 4, 5, 7, 9, 11 | Chapter 10, page 187: | 1, 2, 4, 7 | ||

Exercise 2 | Chapter 10, page 194-195: | 1, 2, 3, 4, 8, 10, 11, 12 | page 202-203: | 1, 2, 3, 4, 7 | ||

Exercise 3 | Chapter 10, page 209: | 1, 2, 4, 6 | page 210: | 1, 2 | ||

Exercise 4 | Chapter 14, page 252-253: | 1, 3, 4, 5, 8, 10, 11 | page 260: | 2, 3, 4, 6, 7 | ||

Exercise 5 | Chapter 14, page 262-263: | 1, 2, 3, 5, 6 | page 268: | 1, 2, 4, 5, 7, 8, 9 | ||

Exercise 6 | Chapter 19,1, page 368: | 1, 2 | 19,2, page 381: | 1, 4, 6 | ||

Exercise 7 | Chapter 19,2, page 381-382 | 2, 7, 8, 9, 11 | ||||

Exercise 8 | Chapter 19,3, page 388 | 3, 5 | chapter 20,3, page 401 | 1,2,3 | ||

Exercise 9 | Exam Dec 11 2007 | Problems 1, 2, 3 | Nov 30 2005 | Problems 1, 4 | ||

Exercise 10 | Dec 11 2007 | Problem 4 | Nov 30 2005 | Problem 2, 3 | Dec 11 2009 | Problem 2, 3 |

Exercise 11 | Dec 13 2010 | Dec 2011 | dec 2012 | dec 2014 | dec 2014 |