Algebra

Code:

M4129

Acronym:

M4129

Level:

400

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2023/2024 - 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	7	Plano Oficial do ano letivo 2021	1	-	9	72	243
			2

Teaching language

English
Obs.: If all students are portuguese, the classes will be in Portuguese

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

Program

(provisonal program)
1.Rings and ideals (nilpotent elements, nil ideals, nil radical, prime ideals, Jacobson radical, operations on ideals)

2. Modules (category of modules, finitely generated modules, exact sequences, tensor product, restriction and extension of scalars, exactness of tensor products, algebras, tensor products of algebras)

3.Rings and Modules of fractions (localization, extension and contraction for ideals in rings of fractions) for commutatve rings.

4. Primary decomposition

5. Integral dependence and valuations (integral dependence, going-up theorem, integrally closed integral domains, going-down, valuation rings)

6. Chain conditions (Noetherian rings, Artinian rings)

7. Discrete Valuation rings and Dedekind domains

8. Semisimple módules, Artin-Wedderburn Theorem.

Mandatory literature

Atiyah M. F.; Introduction to commutative algebra (main literature)
K. R. Goodearl; An introduction to noncommutative Noetherian rings. ISBN: 0-521-36086-2
Donald S. Passman; A course in ring theory. ISBN: 0-534-13776-8

Complementary Bibliography

Eisenbud David; Commutative algebra with a view toward algebraic geometry. ISBN: 0-387-94268-8 (Hardback)
Zariski Oscar; Commutative algebra. ISBN: 0-387-90089-6 (vol. 1)
Matsumura Hideyuki; Commutative algebra. ISBN: 0-8053-7024-2
Reid Miles; Undergraduate commutative algebra. ISBN: 0-521-45255-4
Rodney Y. Sharp; Steps in commutative algebra. ISBN: 0-521-64623-5
I. N. Herstein; Topics in ring theory. ISBN: 0-226-32802-3

Teaching methods and learning activities

2 testes, no final exam

keywords

Physical sciences > Mathematics > Algebra
Physical sciences > Mathematics > Geometry > Algebraic geometry

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	171,00
Frequência das aulas	72,00
Total:	243,00

Eligibility for exams

No restictions

Calculation formula of final grade

Arithemtic mean of the grades of the tests.

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 makeup exame phase.

Observations

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