# Commutative algebra

Course Code : | 1001WETCOA |

Study domain: | Mathematics |

Academic year: | 2017-2018 |

Semester: | 1st semester |

Sequentiality: | - |

Contact hours: | 60 |

Credits: | 6 |

Study load (hours): | 168 |

Contract restrictions: | No contract restriction |

Language of instruction: | English |

Exam period: | exam in the 1st semester |

Lecturer(s) | Kevin De Laet |

### 1. Prerequisites *

At the start of this course the student should have acquired the following competences:

an active knowledge of

an active knowledge of

- English

- English

Knowledge of the course 'Groepen en ringen' ('Groups and rings') and 'Lichamen en Galoistheorie' ('Fields and Galoistheory').

### 2. Learning outcomes *

- The student masters crucial concepts and theorems from Commutative Algebra.
- The student is familiar with various properties of commutative rings and knows diverse natural examples, in particular those which are relevant in algebraic number theory and in algebraic geometry.

### 3. Course contents *

- Commutative rings, ideals, maximal ideals, prime ideals, radical ideals, local rings, Nakayama's Lemma.
- Modules.
- Integral extensions: integral closure, going-up, going-down theorems for chains of prime ideals, Noether Normalization.
- Dimension theory: definition of Krull dimension in relation to the transcendence degree for algebras over a field.
- Noetherian rings: Hilbert's basis theorem, regular local rings.
- Affine algebras: Zariski's Lemma, Hilbert's Nullstellensatz.
- Artinian rings and modules, modules of finite length.
- Dedekind rings and characterization of discrete valuation rings.
- Number fields, applications to problems from algebraic number theory.

### 4 International dimension*

The course has an international dimension.

### 5. Teaching method and planned learning activities

Class contact teachingLectures Practice sessions

Personal workExercises

Personal work

### 6. Assessment method and criteria

The intermediate test and the participation cannot be repeated. In the case where the resit exam is taken, the final grade for the second session is computed in the same way (see 5.3), on the basis of the original grades for participation and the test.

Examination

Continuous assessment

### 7. Study material *

#### 7.1 Required reading

Course notes are provided via blackboard.

**7.2 Optional reading**

The following study material can be studied voluntarily :M.F. Atiyah, I.G. Macdonald: Introduction to Commutative Algebra, Addison-Wesley Publ. Co. Inc., Reading Massachusetts, 1969.

### 8. Contact information *

Karim Johannes Becher (office G.224); email: karimjohannes.becher@untwerpen.be