The first two sections review some fundamental concepts of commutative algebra, while the remaining sections present topics which may not have been seen in an undergraduate introductory course on commutative algebra.
The content of the notes, which has been selected from the material in the references at the end, is intended to prepare the learner to explore the applications of commutative algebra to a broad range of research areas.
Within the lecture notes, there are several exercises at the end of each section.
Every student is encouraged to attempt as many of these as they wish, and more from the selected bibliography.
Your lecturer will recommend some (bi-)weekly exercises, taken from the notes, and the solutions of these will be made available on the MAGIC website by the end of the course.
Every student should be familiar with the relevance of engaging with the practical aspects of a course.
The entire material covered in lectures is examinable, including the exercises. More details about examination will follow in due time.
For the computer-algebra enthusiasts, I included the reference of a resource which may be of interest, and I also would recommend the numerous possibilities offered by MAGMA:
http://magma.maths.usyd.edu.au/magma/.
We do not use any of them in the course, and no assessment component will require any knowledge of software.
Accessibility: please contact your lecturer if you need an alternative format for the lecture notes and exercises.
No prerequisites are absolutely essential, but it helps to have some background in commutative rings and ideal theory.
1. Commutative algebra: the essentials (rings, ideals, homomorphisms, localisation)
2. Modules
3. Integral dependence
4. Prime and maximal ideal spectra
5. A brief taste of algebraic geometry: algebraic sets and Hilbert's Nullstellensatz
6. Primary decomposition
7. Dimension in commutative rings.