Algebraic Structures

Establish the fundamental algebraic structures of mathematics to be used in other disciplines.

To know the basic notions of group theory.

Study the generalisation of the concept of divisibility of rings of integers and rings of polynomials to other rings, using the results to obtain the module structure theorems on these rings.

Know the language of modules over commutative rings.

Obtain the module structure theorems of finite type over domains of principal ideals.

1. GROUPS (11 expositive sessions)
Groups. Subgroups. Normal subgroups. Homomorphisms of groups. Quotient group. The isomorphism theorems. Group action on a set. Symmetries and permutations: the symmetric group. The Cayley theorem. The Sylow theorems.

2. RINGS (10 expositive sessions)
Rings. Subrings. Ideals. Homomorphisms of rings. Quotient rings. The isomorphism theorems. Fields. The characteristic of a field. Prime ideals and maximal ideals. Ideal operations. The chinese remainder theorem. The Jacobson radical. The field of fractions of a domain. Irreducible elements. Unique factorization domains. Principal ideal domains. Euclidean domains. Polynomial rings.

3. MODULES (10 expositive sessions)
Modules. Submodules. Quotient modules. Homomorphisms of modules. The isomorphism theorems. Cyclic modules. Direct products and direct sums of modules. Free modules. Generators and relations.

4. STRUCTURE THEOREMS FOR MODULES OVER A P.I.D. (11 expositive sessions)
Equivalence of matrices. Diagonalization. The structure theorem. Torsion modules and primary components. Invariants

Basic:

Chamizo Lorente, Fernando. Apuntes de Álgebra II. U.A.M. 2005.http://matematicas.uam.es/~fernando.chamizo/libreria/fich/APalgebraII04…
Cohn, P. M. Algebra, Vol. 1 (2ª Ed.). Wiley and Sons, Chichester, 1982.
Jacobson, N. Basic Algebra I, Freeman and Company, San Francisco, 1985.
Lang, S. Algebra. Addison-Wesley, New York, 1993.
Rotman, J. J.Advanced Modern Algebra (2ª Ed.). Prentice Hall, New Jersey, 2003.


Complementary:

Atiyah, M. F., I. G., Macdonald, An Introduction to Commutative Algebra, Addison-Wesley, Massachusetts, 1969.
Gardiner, C. F. A First Course in Group Theory. Springer-Verlag, New York, 1980.
Hartley, B., T. O. Hawkes. Rings, Modules and Linear Algebra.Chapman and Hall, London,1970.
Hilton, P. J., Yel-Chiang Wu. Curso de Álgebra Moderna. Reverté, Barcelona, 1977.
Rotman, J. J. An Introduction to the Theory of Groups.Springer,New York, 1994.

Contribute to achieving the general, specific and cross-cutting competences set out in the USC Degree in Mathematics Degree Report and, in particular, the following (CG3, CG4, CE4, CT1 and CT5):


Apply both the knowledge acquired and the capacity for analysis and abstraction in the formulation of problems and in the search for their solutions.

Written and oral communication of mathematical knowledge, methods, ideas and results.

Identification of errors in incorrect reasoning.

Use of bibliographical resources on the topics of the subject.


Specific competences:

Know and understand the fundamental concepts of group theory and handle them.

Know and understand the fundamental concepts of ring theory and handle them.

Know and understand the fundamental concepts of module theory and handle them.

Lectures will be used to present the basic contents of this subject (CE1, CE2, CE6, CG1, CG4).

In the interactive laboratory classes in very small groups, students will work individually and/or in groups on questions and problems proposed (CB2, CB3, CE3, CE4) and presentations will be given (CB4, CG4).

In the classroom tutorials in very small groups, students' learning and work outside the classroom will be monitored on a personalised basis (CG5, CG4, CT5).

Problem papers will be posted on the virtual course, scheduling them in a staggered manner and always in relation to the theory.

Continuous assessment combined with a final exam is foreseen as an assessment criterion. This final exam will be held on the date set by the Faculty of Mathematics for this purpose.

The continuous assessment will consist of 1 face-to-face test. May not coincide for the different groups but will be coordinated and similar.

The final exam will be the same for both expository groups.

Calculation of the final grade:
The grade for both the first opportunity and the second will be max{F; 0,3xC + 0,7xF} where C denotes the grade for the continuous assessment and F the grade for the corresponding final test.

For cases of fraudulent performance of exercises or tests, the provisions of the Regulations on the evaluation of students' academic performance and revision of grades shall apply.

A No Show will be understood to be a student who does not take the final exam at either the first or the second opportunity.

Classroom work:

Lectures: 42 hours
Laboratory classes: 14 hours
Tutorials in very small groups: 2 hours
Total: 58 hours

Personal work of the student: 92 hours


Total hours of work: 150 hours

Study daily with the help of bibliographic material. Attentively read the theoretical part until it is assimilated and try to answer the questions, exercises or problems presented in the bulletins.