Non-commutative Algebra

Code:

M561

Acronym:

M561

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2018/2019 - 2S

Active?	Yes
Responsible unit:	Department of Mathematics
Course/CS Responsible:	Doctoral Program in Mathematics

Cycles of Study/Courses

Acronym	No. of Students	Study Plan	Curricular Years	Credits UCN	Credits ECTS	Contact hours	Total Time
IUD-M	4	PE do Prog Inter-Univ Dout Mat	1	-	9	60	243

Teaching language

English

Objectives

The program of this course will focus on the structure of non-commutative rings. We will cover basic facts on division rings and quaternion algebras. The construction of the Brauer group of a field and its basic properties will be discussed. A brief outlook to the Brauer group of a commutative ring will be provided. Furthermore, we will discuss Noetherian rings and Goldie's theorem, skew polynomial rings and division rings that are infinite dimensional over their center.

Learning outcomes and competences

The expected outcome is that the student knows some of the recent trends in the theory of (finite or infinite dimensional) Hopf algebra.

Working method

Presencial

Pre-requirements (prior knowledge) and co-requirements (common knowledge)

Linear Algebra and basic abstract algebra, including basic facts on groups, rings and modules.

Program

The syllabus will roughly be as follows:


0. Preliminaries
1. Quaternons and skew fields
2. Tensor products
3. Central algebras over fields
4. Simple algebras
5. The Brauer group of a field
6. Maximal subfields
7. Crossed Products
8. Azumaya Algebras
9. The first Weyl algebra
10. Skew polynomial rings
11. Injective Modules
12. Singular submodules and Goldie's theorem

Mandatory literature

T. Y. Lam; A first course in noncommutative rings. ISBN: 0-387-97523-3
K. R. Goodearl; An introduction to noncommutative Noetherian rings. ISBN: 0-521-36086-2
J. C. McConnell; Noncommutative noetherian rings. ISBN: 0-471-91550-5

Complementary Bibliography

I. N. Herstein; Noncommutative rings

Teaching methods and learning activities

Traditional teaching

Evaluation Type

Distributed evaluation with final exam

Assessment Components

designation	Weight (%)
Exame	50,00
Trabalho escrito	50,00
Total:	100,00

Amount of time allocated to each course unit

designation	Time (hours)
Estudo autónomo	30,00
Frequência das aulas	20,00
Trabalho escrito	50,00
Total:	100,00

Eligibility for exams

Each student has to hand-in two home work assignments and has to pass a final exam. Each homework assignment counts 25%, while the final exam counts 50% of the final grade. There is the chance of improving the grade of the final exam by doing a make-up exam. The grade of the homework assignment cannot be improved. Students are allowed to discuss the homework assignments among themselves, but are not allowed to copy solutions from the internet. Results from textbooks, monographs or research papers, used for the student's solution, have to be clearly identified as such.

Calculation formula of final grade

The final grade is the sum of the 2 written home work assignment  and the final exam.

Classification improvement

Only the grade of the exam can be improved through a make-up exam.