Commutative Algebra

Year
1
Academic year
2023-2024
Code
02021935
Subject Area
Mathematics
Language of Instruction
Portuguese
Mode of Delivery
Face-to-face
Duration
SEMESTRIAL
ECTS Credits
6.0
Type
Elective
Level
2nd Cycle Studies - Mestrado

Recommended Prerequisites

Basic courses in Linear Algebra, Group Theory and Ring and Field Theory.

Teaching Methods

Classes are expository and include examples and exercises for applying the acquired knowledge, where the students

learn by practicing on real cases. The presentation of new concepts is preceded by exemplification or  starts with the abstract concept followed by its ramifications and applications to the concrete. Part of the time is devoted to problem-solving, where the students are encouraged to discuss and work in group.

During the semester students may use tutorial time to clarify their difficulties.

 

Learning Outcomes

To introduce the students to abstract concepts of commutative ring theory and module theory. Development of their capabilities of abstraction and discussion. Identification of fundamental structures of Mathematics, knowledge of their origins and its use in the concrete. Study of the most important properties in ideal and module theories, and their application to geometry.

The course aims at developing the following skills: generalization and abstraction; ability to formulate and solve problems; logic reasoning; independent thinking; research and independent learning; teamwork; imagination and creativity; critical mind; communication skills.

Work Placement(s)

No

Syllabus

Rings: Rings, principal ideals and prime ideals (revisited); principal ideal rings; divisibility and prime factorization; unique factorization domains and Euclidean domains; isomorphism theorems for rings.

Modules: modules over rings, linear independence; modules over integral domains; applications to groups and matrices; chain conditions, noetherian rings and modules; Hilbert’s Basis Theorem.

(Advanced Topics)

Depending on the available time, some advanced topics from commutative ring and module theory can be treated, chosen from the following list: tensor products, the concept of algebraic variety; ideals factorization; maximal ideals and Nakayama’s Lemma; Hilbert’s Nullstellensatz Theorem; decomposition of a variety in a union of irreducible varieties.

Head Lecturer(s)

Alexander Kovacec

Assessment Methods

Assessment
There are 2 types of grading: during the semester or by final examination. a) During the semester there are one mid-term exam (60-75% of the final grade) and two tests (40-25% of the final grade). b) The final exam option consists of a single exam (100% of the final grade). : 100.0%

Bibliography

R. L. Fernandes, M. Ricou, Introdução à Àlgebra, IST Press, 2004.

 

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

 

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