# MA3201 Rings and modules

# News

2.12 As mentioned in the Course description, the exam will be given only in English, but note that answers can be given in either Norwegian or English.

21.11 Solution sheet 6 available below. Office hours for the next few weeks,

Room 834, 8th floor of Sentralbygg II:

Date | Time |
---|---|

Tuesday 25 November | 10-11am |

Thursday 27 November | 10-11am |

Tuesday 2 December | 10-11am |

Thursday 4 December | 10-11am |

Tuesday 9 December | 10-11am |

Thursday 11 December | 10-11am |

Monday 15 December | 10-11am |

Tuesday 16 December | 10-11am |

17.11 Syllabus available below.

17.11 Notes on differential equations added below.

11.11 Problem Sheet 6 and Solution Sheets 4 and 5 available below.

29.10 Change to programme for Friday sessions.

Friday 31 October: Both slots will be used for a lecture.

Friday 7 November: Both slots will be used for Problem Sheet 5.

Friday 14 November: Lecture/Problem sheet 6 (as usual).

Friday 21 November: Lecture/Problem sheet 6 (as usual).

22.10 Problem Sheet 5 available below.

15.10 Solution sheet 3 available below.

08.10 Problem sheet 4 available below.

26.09 Solution sheet 2 available below.

24.09 Problem sheet 3 available below.

14.09 Solution sheet 1 available below.

10.09 Problem sheet 2 available below.

08.09 Sheet containing some key definitions added. Note also past exam papers available below.

05.09 The reference group members are Are Austad, AndrĂ© Prater and Neyah Rizzello. The first reference group meeting will be on Friday 12 September.

# General information

Course description from the student handbook.

Lecturer: http://www.maths.leeds.ac.uk/~marsh, email: marsh [at] maths [dot] leeds [dot] ac [dot] uk, office: Sentralbygg II, room 834, telephone 735 91695.

Office hours: Wednesdays 11:00-12:00, Sentralbygg II, Room 834

Teaching hours:

The second part of the Friday slot will be used for discussion of the problems.

# 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 |

# Plan/log

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

20.08 | Rings and examples | Sections 9.1, 9.2, 9.3 |

22.08 | Examples of rings, algebras over a field, path algebra of a quiver, subrings, centre, characteristic, idempotents, nilpotent elements, direct product of rings, ideals | Sections 9.3, 9.4, 9.5, 10.1; Syllabus sheet |

27.08 | Examples of ideals, ideals in matrix rings, intersection of ideals, ideal generated by a subset, principal ideal, principal ideal domain (PID), quotient ring | Section 10.1 |

29.08 | polynomial ring as PID, quotient ring, examples of quotient rings, homomorphisms, kernel, image, fundamental theorem of homomorphisms | Sections 10.1, 10.2 |

03.09 | Fundamental theorem of homomorphisms, correspondence theorems, examples | Section 10.2 |

05.09 | Sums and direct sums of ideals, relationship to idempotents | Sections 10.3 |

10.09 | Maximal ideals, simple rings, prime ideals | Sections 10.4 |

12.09 | Zorn's Lemma, maximal ideal containing a given ideal | Section 10.6 |

17.09 | A prime ideal is maximal in a PID, modules, examples, submodules, sums of submodules | Sections 10.6, 14.1, 14.2 |

19.09 | Sums of submodules, submodules generated by subsets, linear independence, bases, free modules, module homomorphisms, external direct sum | Sections 14.2, 14.3, 14.5 |

24.09 | Quotient modules, fundamental theorem of R-homomorphisms, correspondence theorem, cyclic modules, Endomorphisms of a ring R regarded as an R-module | Section 14.3 |

26.09 | Endomorphisms of a ring R regarded as an R-module, simple modules | Section 14.3 |

1.10 | Schur's lemma, Semisimple modules (completely reducible modules), writing as direct sums, noetherian modules, noetherian rings | Sections 14.4, 19.2 |

3.10 | Characterization of noetherian modules; given a submodule, a module is noetherian if and only if a given submodule and its quotient are noetherian; Artinian modules | Section 19.2 |

8.10 | Artinian modules; artinian rings; characterization of artinian modules; a module is artinian if and only a given submodule and its quotient are artinian; a direct sum of artinian modules is artinian; a direct product of left artinian rings is left artinian; a nil left ideal in an artinian ring is nilpotent; endomorphism ring of a direct sum of modules | Sections 19.1, 19.2 |

10.10 | Wedderburn-Artin theorem - proof using key lemma | Section 19.3 |

15.10 | Wedderburn-Artin theorem - proof of key lemma; Extended version of Wedderburn-Artin theorem | Section 19.3 |

17.10 | Comments on Wedderburn-Artin theorem; prime and irreducible elements in PIDs | Section 11.1 |

22.10 | prime and irreducible elements in PIDs; associates; unique factorization domains (UFDs); Every PID is a UFD; Greatest common divisors in a UFD; uniqueness up to associates; expressing gcd(a,b) in form sa+tb; rank of a free module over a commutative ring; rank of a submodule of a free module over a PID; row-module, row rank, column-module and column rank of a matrix over a PID; equivalent matrices; row-rank and column rank are the same for equivalent matices; elementary row and column operations; | Sections 11.1, 11.2, 14.5, 20.1, 20.2, 20.3 |

24.10 | Non-elementary row and column operations; Smith normal form; Theorem: every matrix over a PID is equivalent to one in Smith normal form | Section 20.3 |

29.10 | Proof of theorem that every matrix over a PID is equivalent to one in Smith normal form; corollary that the row rank and column rank of a rectangular matrix over a ring coincide; discussion of Smith normal form in the case of Z and F[x]; examples | Section 20.3; Syllabus sheet |

31.10 | Structure theorem and proof for finitely generated modules over a PID; statement of uniqueness; application of structure theorem to the classification of finitely generated abelian groups; Introduction to rational canonical form | Sections 21.1, 21.2, 21.3, 21.4 |

5.11 | F-vector space with linear transformation with matrix A as module over F[x] in terms of invariant factors of A-xI; Rational canonical form of a square matrix over a field; examples; characteristic and minimal polynomials and relationship to invariant factors; | Section 21.4; Syllabus sheet |

7.11 | Discussion of Problem sheet 5 | |

12.11 | Minimum polynomial divides characteristic polynomial; Cayley Hamilton Theorem; invariant factor examples; Jordan canonical form: elementary divisors, examples | Section 21.5 |

14.11 | Jordan canonical form and examples | Section 21.5 |

19.11 | Revision | |

21.11 | Revision (and more discussion of Problem Sheet 6) |

# Exam

Final exam: 17.12.2013, written, 4 hours, 09:00-13:00

# Problem sessions

The second hour of the Friday slot will be used for discussion of the problems.

Dates | Problem sheet | Solution sheet |
---|---|---|

05.09, 12.09 | Problem sheet 1 | Solution sheet 1 |

19.09, 26.09 | Problem sheet 2 | Solution sheet 2 |

03.10, 10.10 | Problem sheet 3 | Solution sheet 3 |

17.10, 24.10 | Problem sheet 4 | Solution sheet 4 |

07.11 only | Problem sheet 5 | Solution sheet 5 |

14.11, 21.11 | Problem sheet 6 | Solution sheet 6 |

# Reference group

Reference group members |
---|

Are Austad |

AndrĂ© Prater |

Neyah Rizzello |

Dates of meetings |
---|

12 September 2014 |

10 October 2014 |

14 November 2014 |