Algebra

Code:

M4075

Acronym:

M4075

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2020/2021 - 1S

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

Cycles of Study/Courses

Acronym	No. of Students	Study Plan	Curricular Years	Credits UCN	Credits ECTS	Contact hours	Total Time
M:M	8	Plano de Estudos do M:Matemática	1	-	6	56	162

Teaching language

Portuguese and english

Objectives

The student should know and understand the concepts and basic results of the theory of rings and modules, including basic familiarity with the classical examples. It is intended that this unit contribute to the development of skills of abstract reasoning and familiarity with the mathematical method.

Learning outcomes and competences

Familiarity with the main results and techniques of basic ring theory.

Working method

Presencial

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

Prerequisites: two semesters of linear algebra, one semester of group theory.

Program

This is an introductory course in ring theory with an emphasis on modules. We will study several important classes of rings and modules, bringing forth aspects of commutative and noncommutative ring theory. Both major classical results as well as recent directions of research will be highlighted.

The following items will be covered:

1) Rings: Fields, division rings, matrix rings, semigroup rings, free rings

2) The category of modules: Modules and homomorphisms. Endomorphism rings. Simple modules, finitely generated modules and free modules. Sum of modules (internal and external).

3) Rings and modules with chain conditions. Differential operator rings.

4) Finite length modules: the Jordan-Hölder Theorem.

5) Projective and injective modules.

6) The Artin-Wedderburn theory: semisimplicity, structure of semisimple rings.

7) The Brauer group of a division ring

8) The Jacobson Radical and the density theorem.

9) Modules over finite dimensional algebras.

Mandatory literature

Passman Donald S.; A course in ring theory. ISBN: 0-534-13776-8
Lam T. Y.; A first course in noncommutative rings. ISBN: 0-387-97523-3

Complementary Bibliography

Brec5a1ar Matej; Introduction to noncommutative algebra. ISBN: 9783319086927
Goodearl K. R.; An introduction to noncommutative Noetherian rings. ISBN: 0-521-36086-2
Rowen, Louis; Ring Theory (students edition), Academic Press, Inc., 1991. ISBN: 0-12-599840-6
Herstein I. N.; Topics in ring theory. ISBN: 0-226-32802-3

Teaching methods and learning activities

The contents of the syllabus are presented in the lectures, where examples are given to illustrate the concepts.

keywords

Physical sciences > Mathematics > Algebra

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	112,00
Frequência das aulas	56,00
Total:	168,00

Eligibility for exams

Without restrictions.

Calculation formula of final grade

Student evaluation is based on the individual resolution of exercise sets and a final exam.

The final grade is the average of the grades obtained in the exercise sets (50%) and the final exam (50%). 

Due to the current pandemic situation and as a consequence of the last decission of the "conselho pedagógica", the evaiuation for the second evaluation round ("recurso") will be based on an homework assignment which the students have to hand-in within 24 hours.

Special assessment (TE, DA, ...)

Special exams will consist of a written test, which might be preceded by an eliminatory oral test to assess whether the student satisfies minimum requirements to tentatively pass the written test.

Classification improvement

Due to the current pandemic situation and as a consequence of the last decission of the "conselho pedagógica", the evaiuation for the second evaluation round ("recurso") will be based on an homework assignment which the students have to hand-in within 24 hours.

The distributed part of the assessment (all except for the final exam) cannot be retaken. The students trying to improve the scores from the previous year (2019/2020) will have also to do homework similar and will have to hand it in until 4 days before the last day of classes.

Observations

Any student may be required to take an oral examination should there be any doubts concerning his/her performance on certain assessment pieces.