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Fundamental Algebra
| Code: | M501 | Acronym: | M501 |
| Keywords | |
|---|---|
| Classification | Keyword |
| OFICIAL | Mathematics |
Instance: 2024/2025 - 1S 
| 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 | 2 | PE do Prog Inter-Univ Dout Mat | 1 | - | 9 | 60 | 243 |
Teaching language
PortugueseObjectives
Introduction to fundamental topics in abstract algebra.Learning outcomes and competences
Familiarity with basic concepts and results of abstract algebra (rings, modules).Working method
PresencialProgram
he program is centered on the theory of rings (not necessarily commutative) and modules
1) Rings and ideals: rings, ring homomorphisms, ideals and quotient rings, other elementary properties of rings
2) Modules: basic concepts, exact sequences, direct sums and products, internal direct sum, free modules
3) Projective and injective modules: the group of homomorphisms, projective modules, injective modules
4) Some Structure Theorems: ordered sets, chain conditions, Zorn's Lemma, Noetherian rings and Artinian rings, composition sequences and the Jordan-Holder Theorem for modules, the Fitting Lemma, the Krull Schmidt Theorem, semisimple modules, simple and semisimple rings, the Artin-Wedderburn Theorem
Other topics may be covered depending on time availability (such as Jacobson's radical and Nakayama's Lemma) and student interest.
1) Rings and ideals: rings, ring homomorphisms, ideals and quotient rings, other elementary properties of rings
2) Modules: basic concepts, exact sequences, direct sums and products, internal direct sum, free modules
3) Projective and injective modules: the group of homomorphisms, projective modules, injective modules
4) Some Structure Theorems: ordered sets, chain conditions, Zorn's Lemma, Noetherian rings and Artinian rings, composition sequences and the Jordan-Holder Theorem for modules, the Fitting Lemma, the Krull Schmidt Theorem, semisimple modules, simple and semisimple rings, the Artin-Wedderburn Theorem
Other topics may be covered depending on time availability (such as Jacobson's radical and Nakayama's Lemma) and student interest.
Mandatory literature
Joseph J. Rotman; Advanced modern algebra. ISBN: 978-1-4704-1554-9 : 1a pt.Rotman, J.J.; Advanced Modern algebra, ams, 2010. ISBN: 978-0-8218-4741-1
César Polcino Milies; Anéis e Módulos, Editora Livraria da Física 2a edição, 2018
Teaching methods and learning activities
The course material is presented and developed in the lectures, where exercises are also discussed.Evaluation Type
Distributed evaluation without final examAssessment Components
| designation | Weight (%) |
|---|---|
| Apresentação/discussão de um trabalho científico | 50,00 |
| Trabalho escrito | 50,00 |
| Total: | 100,00 |
Amount of time allocated to each course unit
| designation | Time (hours) |
|---|---|
| Frequência das aulas | 60,00 |
| Total: | 60,00 |
Eligibility for exams
Course registration is the only requirement.Calculation formula of final grade
Assessment of written work and respective oral presentation.
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Page created on: 2025-11-20 at 12:44:22 | Privacy Policy | Personal Data Protection Policy | Whistleblowing | Electronic Yellow Book
Page created on: 2025-11-20 at 12:44:22 | Privacy Policy | Personal Data Protection Policy | Whistleblowing | Electronic Yellow Book