2020/2021

Degree | Type | Year | Semester |
---|---|---|---|

2500149 Mathematics | OT | 4 | 0 |

The proposed teaching and assessment methodology that appear in the guide may be subject to changes as a result of the restrictions to face-to-face class attendance imposed by the health authorities.

- Name:
- Ferran Cedó Giné
- Email:
- Ferran.Cedo@uab.cat

- Principal working language:
- catalan (cat)
- Some groups entirely in English:
- No
- Some groups entirely in Catalan:
- Yes
- Some groups entirely in Spanish:
- No

- Laurent Cantier

The knowlege of all the compulsory courses in algebra, specially the course *Estructures Algebraiques*.

The aim of this course is to give an introduction a to the commutative ring theory.

This means to work with the basic concepts of commutative rings, their ideal structure and the modules over them.

At the end of this course, the students should know localization in rings and modules, factorization in commutative rings and integral dependence.

- Actively demonstrate high concern for quality when defending or presenting the conclusions of one’s work.
- Assimilate the definition of new mathematical objects, relate them with other contents and deduce their properties.
- Demonstrate a high capacity for abstraction.
- Develop critical thinking and reasoning and know how to communicate it effectively, both in one’s own languages and in a third language.
- Effectively use bibliographies and electronic resources to obtain information.
- Students must be capable of applying their knowledge to their work or vocation in a professional way and they should have building arguments and problem resolution skills within their area of study.
- Students must be capable of collecting and interpreting relevant data (usually within their area of study) in order to make statements that reflect social, scientific or ethical relevant issues.
- Students must be capable of communicating information, ideas, problems and solutions to both specialised and non-specialised audiences.
- Students must develop the necessary learning skills to undertake further training with a high degree of autonomy.

- Actively demonstrate high concern for quality when defending or presenting the conclusions of one’s work.
- Develop critical thinking and reasoning and know how to communicate it effectively, both in one’s own languages and in a third language.
- Effectively use bibliographies and electronic resources to obtain information.
- Students must be capable of applying their knowledge to their work or vocation in a professional way and they should have building arguments and problem resolution skills within their area of study.
- Students must be capable of collecting and interpreting relevant data (usually within their area of study) in order to make statements that reflect social, scientific or ethical relevant issues.
- Students must be capable of communicating information, ideas, problems and solutions to both specialised and non-specialised audiences.
- Students must develop the necessary learning skills to undertake further training with a high degree of autonomy.
- Understand in-depth demonstrations of some theorems of advanced algebra and assimilate the definition of new algebraic structures and constructions, relating them with other knowledge and deducing their properties.
- Use algebraic tools in different fields.

This course will follow essentialy the book "M.F. Atiyah i I.G. Macdonald, Introducción al Álgebra Conmutativa".

The contents will be:

1. Commutative rings.

2. Module theory.

3. Chain conditions.

4. Integral dependence.

5. Dimension theory.

The realization of parts 4 and 5 will depend on the development of the course.

This course has two hours of theory par week. The recommended bibliography is interesting; sometime during the course the students should complement and complete the content of the lectures using this bibliography.

There are problem classes (one hour par week). Every student should present the solutions of some lists of problems on the blackboard or in paper to the professor. To solve doubts about the course the student can ask questions to the professors during the class or in the office (with some schedule of times). The solution of these problems will be based in general on the theory: the theorems and their proofs.

The 6 hours of seminar classes of this course will be dedicated to solve problems collaborating several groups of students in the class room.

The course has a web “campus virtual” where the professors will put the lists of problems and other information about the course.

Title | Hours | ECTS | Learning Outcomes |
---|---|---|---|

Type: Directed | |||

Theory classes | 30 | 1.2 | |

Type: Supervised | |||

Problem classes | 15 | 0.6 | |

Seminars | 6 | 0.24 | |

Type: Autonomous | |||

Solving problems | 60 | 2.4 | |

Study of the theory | 36 | 1.44 |

The evaluation of the course will be a 50% of continued evaluation, and a 50% of exams.

- The continued evaluation will consist on the presentation of solutions of problems (solved individually), the attendance to the seminars and the presentation of solutions of the exercises proposed in the seminars (solved in group).
- The mark of the exams will be obtained from a final exam.

The "matrícules d'honor" will be decided taken into account the results of the continued evaluation and the exams.

Title | Weighting | Hours | ECTS | Learning Outcomes |
---|---|---|---|---|

Final exam | 50% | 3 | 0.12 | 8, 2, 7, 6, 4, 5 |

Seminar attendance | 10% | 0 | 0 | 1, 2, 6, 4, 5, 3, 9 |

Solved problems | 40% | 0 | 0 | 1, 2, 6, 4, 3, 9 |

W. A. Adkins, S. H. Weintraub, Algebra, An Approach via Module Theory. Springer, New York, 1992.

A. Altman, S. Kleiman, A Term of Commutative Algebra. Worldwide Center of Mathematics, LLC, 2012.

M. Atiyah, I. Macdonald, Introducción al álgebra conmutativa. Ed. Reverté, Barcelona, 1968.

P. M. Cohn, Algebra, vol 2. Second Ed. John Wiley and Sons, New York, 1989.

D. Eisenbud, Commutative Algebra with a view toward Algebraic Geometry. Springer, New York, 2004.

B. Hartley, T. O . Hawkes, Rings, modules and linear algebra. Chapman and Hall, London 1983.

N. Jacobson, Basic Algebra I, Basic Algebra II. W. H. Freeman and Company, New York, 1989.

E. Kunz, Introduction to Commutative Algebra and Algebraic Geometry. Birkhäuser, New York, 2013.

S. Lang, Algebra. Aguilar, Madrid, 1977.

J.S. Milne, A Primer of Commutative Algebra, 2009.

O. Zariski, P. Samuel, Commutative Algebra I, II, Van Nostrand, Princeton (1958, 1960).