# MA3201 Rings and modules

# News

14.08: First lecture Thursday.20/8, at 12:15 in K28 (likker i Kjemiblokk 1, 2.etg).

# General information

**Lecturer:** sverre o. smalø, email: sverresm at math ntnu no, SBII, room 850, tlf. 735 91750

**Office hours:** Thursdays 14:15-15:00, SBII, room 850. I am awailable all days between 9.00 to 15.00 from now on and until December 14th. (Only by appointment for Saturdays and Sundays)

Teaching hours:

Lectures: | Thursdays 12:15-14:00, K28 |

Friday 14:15-16:00, MA24 |

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

The problem session on Friday October 2nd will be given by Markus Schidmeier, with problems from from Exercises 2-6.

The problem session on Thursday 22nd of October will be given by Magnus Hellstrøm-Finnsen. The problem discussed will be from Exercises 7-8 + one given in class.

Thursday October 29th: The Jordan-Hölder theorem for modules which are both Noetherian and Artinian was given.

Friday October 30th: The exam problem 1 and 2 from dec. 11th 2007 was given together with problem 4 from the exam November 30th 2005.

Thursday Nov. 5th: The plan is to do problem 4 from Dec. 11th, 2007; and 1, 2 and 3 from Nov.30th, 2005.

Friday Nov. 6th: Problems from Dec. 2010 and Dec. 2011.

# 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 |
---|---|---|

19.08 | Introduction, definitions, various examples | Chap 9.1, 9.2, 9.3 |

23.08 | Polynomial rings and formal power series, subrings, characteristic of a ring, algebras, ideas and homomorphisms | Chap 9.4, 9.5, 10.1, 10.2 |

26.08 | More on Ideals and homomorphisms, left ideals, right ideals, ideals in the full matrix algebra, sums and direct sums of ideals, products of rings, | Chap 10.1, 10.2, 10.3 |

30.08 | Minimal, maximal and prime ideals, the real numbers as the rings of Cauchy-sequences modulo the ideal of sequences converging to zero | Chap 10.4 |

02.09 | Maximal ideals are prime, the maximal ideals of Z, C[x] and R[X], description of prime ideals for commutative rings. Nilpotent and nil ideals, Zorns's lemma and the existence of maximal ideals. | Chap 10.5,10.6 |

06.09 | Modules and Vectorspaces | Chap 14.1, 14.2 |

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

09.13 | Some exercises, the rest of 14.3 and started on 14.4 | Chap 14.3, 14.4 |

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

09.20 | Finished 14.4 started on 19.2 | Chap 14.5, 19.2 |

09.23 | Finished therem 19.2.5 and 19.2.6, gave examples, and finished section 19.1 | Chap 19.1, 19.2 |

09.27 | We will take a look at the exercises, so look through the exercises given as exercise 1, 2 and 3, and make and efford to try to solve those problems | |

09.30 | 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 |

10.04 | Exercise 2 and 3 | |

10.07 | Th 19.2.11, 19.2.12, 19.2.14, Examples 2.15 a, and b. | Chap 19.2 |

10.11 | Exercise 4 and 5 | |

10.14 | Wedderburn Artin theorem | Chap 19.3 |

10.18 | Exercise 5 and 6 | |

10.21 | Comments on Weddernburn Artin theorem and start on Smith normal form | Chapter 20.1 |

10.25 | Exercise 6 and 7 | |

10.28 | Smith normal form | Chapter 20.1 |

# Exam

Final exam: 15.12.2015, written, 4 hours, 9:00-13:00

# 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, page 368: | 1, 2 | page 381: | 1, 4, 6 | ||

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

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