Commutative Algebra

First Semester 2009

Pure Mathematics Honours Course in the Access Grid Room

convened by David Easdown

This page will be updated periodically through the semester.

The assessment will comprise three assignments, due early April, mid May and late June respectively, and a peer review of the first assignment. (Peer reviews of the second and third assignments were planned, but had to be abandoned due to lack of time.) The third assignment is in the nature of a take-home examination over an extended period of time.

First Assignment due Wednesday 1 April 2009

Second Assignment due Friday 22 May 2009

Third Assignment due Friday 26 June 2009

Contents and overview of core materials

Rings and Ideals

Rings and Ideals continued

Posets and Zorn's Lemma

Local Rings and Residue Fields

Divisibility and Factorisation

The Nil and Jacobson Radicals

Operations on Ideals

Operations on Ideals continued

The Radical of an Ideal

Extension and Contraction

Extension and Contraction continued

Modules and Module Homomorphisms

Modules and Module Homomorphisms continued

Submodules and Quotient Modules

Operations on Submodules

Operations on Submodules continued

Finitely Generated and Free Modules

Nakayama's Lemma

Exact Sequences

Exact Sequences continued

Split Extensions

Snake Lemma

Introducing Tensor Products

Introducing Tensor Products continued

Tensor Products

Tensor Products continued

Multilinear Mappings and Tensor Products

Hint for last question on the second Assignment

Introducing Restriction and Extension of Scalars

Restriction and Extension of Scalars

Introducing Exactness Properties of Tensor Products

Exactness Properties of the Tensor Product

Introducing Algebras


Introducing Rings of Fractions

Rings of Fractions

Introducing Modules of Fractions

Connection With Tensor Products

Modules of Fractions

Introducing the Fraction Functor

Modules of Fractions continued

Introducing Properties of Localisation

Some Properties of Localization

Extended and Contracted Ideals in Rings of Fractions

Extended and Contracted Ideals in Rings of Fractions continued

Introducing Primary Decompositions

Exploring Primaries, Radicals, Maximals and Powers

Primary Decompositions

Chain Conditions

Introducing Jordan-Holder

Composition Series

Introducing Noetherian Rings

Background on Chain Conditions

Introducing Hilbert's Basis Theorem

Noetherian Rings

Hilbert's Nullstellensatz (Zeros Theorem)

Appendix: Gauss' Theorem