Commutative rings

You met commutative rings for the first time in Algebra II. In this project, we will pick up where you left off, with a particular focus on rings of polynomials in several variables. For example, we will prove that any ideal of \(\mathbb{C}[x,y,z]\) is finitely generated, and we will classify the maximal ideals of this ring --- which will prove an amazing generalisation of the fundamental theorem of algebra to several variables. We will also study fundamental concepts of localisation, of modules ("vector spaces over a ring"), and of tensor products.

There are almost infinitely many directions the project could go in. You could study algebraic geometry, in which ring theory plays a similar role to that played by analysis in the theory of manifolds. Or you could study computational methods for working with rings; for instance, how can you tell if an ideal is prime? Or you could delve into the different notions of the dimension of a ring. Or the properties of rings of power series. Or even what happens if your rings are (gasp!) noncommutative.

Prerequisites and Corequisites

Algebra II. Would be good if you enjoyed Number Theory III or Galois Theory III, or are taking Representation Theory IV.

Sources:

• Atiyah and MacDonald, "Introduction to Commutative Algebra".
• Cox, Little and O'Shea, "Ideals, Varieties and Algorithms".
• Dummit and Foote, "Abstract Algebra".