**Prerequisites**

A firm grasp of (commutative) rings, ideals, prime ideals and maximal ideals, zero divisors, quotient rings, subrings and homomorphisms, polynomial rings in several variables, finite field extensions, and algebraically closed fields. This material is contained in many standard books on algebra, for example in Chapters 7, 8, 9 (except 9.6), and 13 (except 13.3 and 13.6) and parts of 14.9 of the book 'Abstract algebra' by Dummit and Foote (third edition), or in the book 'Algebra' by Serge Lang (parts of Chapters 2, 3, 5, and 7 will be needed).

The `Intensive Course on Categories and Modules' contains important background material, and should be watched by all students not already familiar with it. The course can be accessed through https://elo.mastermath.nl/

**Aims of the course**

Commutative algebra is the study of commutative rings and their modules, both as a topic in its own right and as preparation for algebraic geometry, number theory, and applications of these. We shall treat the general theory, but also consider how to do explicit calculations.

We hope to cover the following topics:

- flatness

- Nakayama’s lemma

- localisation of rings and modules

- Zariski topology and support

- integral extensions

- Nullstellensatz

- Noetherian and Artin rings and their modules

- associated primes and primary decomposition

- dimension theory and the dimension of fibres

- local rings and regularity

- completions

Other topics may be covered as well.

**Rules about Homework/Exam**

There will be homework sets (counting for 20% total) and a written exam for the remaining 80%. The homework grade will also count for the retake, in the same ratio. The retake exam will likely also be written, but may be an oral exam if very few students need to take it. A student should get at least a 5.0 on the final exam to pass the course (the same holds for the retake, if applicable).

**Literature**

Course book: `Introduction to Commutative Algebra’ by Atiyah-Macdonald

Alternative literature: Commutative Algebra’ by Eisenbud, A Term of Commutative Algebra’ by Altman-Kleiman, Commutative Algebra’ andCommutative Ring Theory’ by Matsumura.

**Lecturers**

David Holmes (UL) & Marta Pieropan (UU)

- Docent: David Holmes
- Docent: Marta Pieropan