Algebra

Code:

M4129

Acronym:

M4129

Level:

400

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2025/2026 - 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	13	Official study plan since 2024/2025	1	-	9	63	243

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

The aim of this course is that students will learn the basic concepts of algebra at the level of a master course.

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. Notions of Ring Theory

2. Commutative Rings

3. Module Theory

4. Goldie theorem 

Mandatory literature

Lam T. Y.; A first course in noncommutative rings. ISBN: 0-387-97523-3
Goodearl K. R.; An introduction to noncommutative Noetherian rings. ISBN: 0-521-36086-2
Passman Donald S.; A course in ring theory. ISBN: 0-534-13776-8

Complementary Bibliography

Brec5a1ar Matej; Introduction to noncommutative algebra. ISBN: 9783319086927
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
John A. Beachy; Introductory lectures on rings and modules. ISBN: 0-521-64340-6

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 without final exam

Assessment Components

designation	Weight (%)
Teste	100,00
Total:	100,00

Amount of time allocated to each course unit

designation	Time (hours)
Estudo autónomo	180,00
Frequência das aulas	63,00
Total:	243,00

Eligibility for exams

Attending classes is not compulsory.

Calculation formula of final grade

There will be 2 testes, the fisrt will during classes and the second will be on the day of the first final exam. On the day of the first final exam, student are allowed to repeat the first test. The final score will be given by the following formula

0.4*T1+0.6*T2

where T1 is the score of the first test and T2 the score of the second.

The second final exam will be divided into 2 parts, each corresponding to one of the tests. Students will have the possibility of writing just one part and having in the other part the score of the correspondent test, if they want.

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

Students that had passed the course in the current or in previous academic years can only improve their grade by taking the exame of the make up exame phase, they are allowed, in case they are students of this academic year, to replace part of the exam by one of the tests.

Observations

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