Commutative Algebra

Recommended Prerequisites

Linear Algebra and Analytic Geometry I, II;

Groups and Symmetries;

Fields and Algebraic Equations.

Teaching Methods

The lessons are mixed in nature, providing students with an opportunity to learn using concrete cases. The concepts are preceded by persistent exemplification. Sometimes the lessons take a slightly opposite turn, beginning with a discussion about an abstract definition and then providing the consequences and the concrete applications of that definition. A substantial part of the lessons is put aside for the solution of problems, where students are encouraged to work and discuss as a group.

Learning Outcomes

Introduction to abstract concepts in commutative rings theory. Development of the capacity for abstraction and argumentation. Acknowledgment of the fundamental structures of Mathematics, as well as their origin and their concrete use. Study of the most important proprieties related to the concepts of ideals and modules, and their application to geometry.

 

Generic competences:

Capacity for generalization and abstraction;

Capacity for formulating and solving problems;

Logical argumentation;

Rigorous and clear written and spoken expression;

Individual initiative;

Capacity for teamwork;

Capacity for autonomous learning;

Imagination and creativity;

Critical thinking;

Communication skills.

Work Placement(s)

No

Syllabus

Main ideals and prime ideals; Special rings and algebras (Boole, fraction, principal ideals). Introduction to module theory. Chain conditions, Noetherian and Artinian rings and modules. The case of polynomial rings. Hilbert and Noether base theorem. The concept of algebraic variety. Decomposition of a variety in a union of irreducible varieties.

Assessment Methods

Assessment
Synthesis work: 10.0%
Resolution Problems: 20.0%
Exam: 70.0%

Bibliography

M. Atiyah & I. MacDonald (1969). Introduction to Commutative Algebra, Add. Wesley.

 

M. Reid (1998-2001). Undergraduate Algebraic Geometry: London Math. Soc. Student Texts 12. Cambridge University Press.