The aim of this course is to provide students with basic tools in Commutative Algebra, which consist essentially in the study of commutative rings, as the ring of integers Z and polynomial rings.
Basic Algebra (groups, rings, fields).
Course unit content
Rings and Ideals.
Modules and exact sequences. Noetherian Rings, Artinian Rings and Dedekind Domains.
Rings and Ring Homomorphisms; Ideals and Quotient Rings; Prime Ideals and Maximal Ideals.
Modules and submodules, operations on submodules; finitely generated modules, homomorphisms and exact sequences.
Noetherian Rings, Artinian Rings and Dedekind domains: definition and properties, Going-Up Theorem and Going-Down Theorems; chain conditions. Hilbert Basis Theorem.
The course is based on frontal lectures. In the lectures we shall propose formal definitions and proofs, with significant examples and several exercises. Exercises are useful for students, to learn how to apply their knowledge to particular cases.
Assessment methods and criteria
Learning is checked in a classic way, through an oral interview.
In the colloquium, students must be able to prove properties of the studied structures, using an appropriate algebraic language and a proper mathematical formalism. In addition, students will be required to apply their knowledge to particular cases.