Commutative Algebra

Credits

Prerequisites

None.

Scientific-disciplinary sector (SSD)

MAT/02 Algebra.

Examination method

Oral exam.

Learning
objectives

The course aims to introduce the fundamental methods and contents of the theory of (commutative) rings, modules and algebras and its applications, with reference to languages and methods also used in other areas of mathematics.

Syllabus

Constructions for rings, modules and algebras. Free modules and algebras, polynomial rings and formal power series rings. Radicals. Sums, products, intersections and quotients of ideals. Nilradical and Jacobson radical. Local rings. Rings of fractions and localisations; extension and contraction of ideals. Nakayama’s lemma; Krull’s intersection theorem. Primary decomposition of ideals. Chain conditions for rings and modules. Extensions of modules. Hom functors. Projective modules. Fractional ideals; Dedekind rings. Bézout rings, valuation rings, integral elements over a ring. Rings of integers in number fields.

Expected learning
outcomes

By the end of the course, students must demonstrate that they

know and understand the topics covered in ring theory, module theory and the theory of algebras
are able to apply the knowledge acquired to connect abstract structures with the related concrete examples, and are able to illustrate the results and computational techniques acquired;
are able to communicate ideas and solutions clearly, rigorously and effectively to both specialist and non-specialist audiences;
are able to identify the most appropriate methods to analyse and solve a problem relating to the course topics and to interpret the results correctly.

Learning outcomes
to be assessed

Command of the knowledge acquired, clarity of presentation, rigour in the use of language, confidence in using the notions acquired.

Commutative Algebra