In this course, the algebraic structures of commutative ring, algebra and module are studied. The main properties of these structures will be studied, mainly concentrating in factorization properties. Thus, special importance will be given to unique factorization domains and, in particular, to rings of polynomials over a field. On the other hand, applications to different parts of algebra will be seen, especially in the case of modules over principal ideal domains.



This course, together with “Algebraic Structures” and “Algebraic Equations”, forms a module in which the fundamentals of abstract algebras and its main applications are developed. The student will obtain the basic techniques of this area, which can be used in other areas of mathematics, and, if they wish, they will be able to continue a deeper study of algebra via the optative fourth-year courses.

SPECIFIC LEARNING RESULTS



M01CM04 Understand the basic concepts of ring and field (subrings, ideals, quotients, homomorphisms, characteristic, field of fractions ...)

M01CM05 Understand the divisibility properties of polynomials in one or more variables and, in particular, learn of apply the main criteria of irreducibility.

M01CM06 Know how to construct and how to use Gröbner bases of ideals of polynomials in several variables to, for example, decide if a polynomial is in an ideal or to delete variables in polynomial systems of equations.

M01CM07 Know the main types of commutative rings (integers, unique factorization domains, Euclidean domains and PIDs) and the relations among them.

M01CM08 Know the basic concepts of module theory over rings.

M01CM09 Understand the structure theorem of finitely generated modules over PIDs amd its applications (Jordan canonical form and Smith normal form).



GENERAL LEARNING RESULTS



Understand the basic concepts of ring theory and, in particular, the theory of polynomial rings in one or more variables.

Understand the structure theorem of finitely generated modules over PIDs amd its applications (Jordan canonical form and Smith normal form).

1. GENERALITIES FOR RINGS: Rings and subrings. Ideals and quotient rings. Homomorphisms and isomorphisms.

2. DIVISIBILITY AND FACTORIZATION: Unique factorization domains. Principal ideals domains (PIDs). Euclidean domains. Applications: some classical arithmetical theorems.

3. POLYNOMIALS IN SEVERAL VARIABLES: Gauss Lemma. Factorization in polynomial rings. Irreducibility criteria.

4. GRÖBNER BASES: Monomial orders in the polynomial ring and division algorithm. Hilbert basis theorem. Basic properties of Gröbner bases. Buchberger algorithm. Applications.

5. MODULES: Modules, first properties and examples. Submodules, quotient modules. Module homomorphisms. Direct sum. Free modules.

6. MODULES OVER PIDs: Annihilators and primary decomposition. Structure theorem for modules over PIDs. Matrices over PIDs: Smith's normal form. Applications: diofantiquee linear equations, finitely generated abelian groups and Jordan canonical form.

The theoretical content will be exposed in theory classes following the basic references and the compulsory material appearing in the bibliography. These theory classes will be complemented by problem sessions, in which the students will be asked to solve problems to apply the knowledge obtained in the theory classes. In the seminars, relevant questions and examples will be discussed directly by the students through exercises, which will be assigned in advance. On the day of the seminar the proposed solutions will be discussed in a critical manner. Additionally, group problems will be proposed to improve teamwork. The solutions to these problems (both seminars and group problems) will be handed in, to be evaluated by the teacher.

An important part of the student's work is individual. Throughout the course, the teachers will guide this individual work and they will stimulate its regularity and the student's dedication. At the same time, the importance of personal tutoring session will be underlined.

• Continuous Assessment System
• Final Assessment System
• Tools and qualification percentages:
  • Ver ORIENTACIONES (%): 100

ORDINARY CALL



The final grade will be obtained via arithmetical mean of the following grades:



O1. Final written exam: 70%-100%

O2. Partial written exam: 0%-10%

O3. Individual problems and/or works (including seminars attendance): 0%-10%

O4. Group problems: 0%-10%



To pass the course, the minimum mark in the final written exam has to be greater than 4,5 out of 10.

The attendance to seminars of compulsory, unless properly justified by a proper supporting document.

In the case that sanitary conditions would obstruct the evaluation activities above specified, for all or part of enrolled students, the corresponding directives of the Rector will be followed.

EXTRAORDINARY CALL



Written exam of the extraordinary call: 100%



For the corresponding students, the totality of the grade will be determined by the written exam of the extraordinary call. Therefore, a mark of 5 in said exam will be necessary to pass the course.

In the case that sanitary conditions would obstruct the evaluation activities above specified, for all or part of enrolled students, the corresponding directives of the Rector will be followed.

Class notes. Proposed problems and exercises.

Basic bibliography

- M.F. ATIYAH, I.G. MACDONALD. Introducción al Álgebra Conmutativa. Reverté, 1973.



- P. CAMERON. Introduction to algebra. Oxford University Press, segunda edición, 2008.



- D. COX, J. LITTLE, D. O'SHEA. Ideals, Varieties and Algorithms. Springer, segunda edición, 1997.



- G. NAVARRO. Un Curso de Algebra. Universitad de Valencia, 2002.

In-depth bibliography

- N. JACOBSON. Basic Algebra. W.H. Freeman and Company, 1985.

- S. LANG. Undergraduate algebra. Springer, tercera edición, 2005.

- M. REID. Undergraduate Conmutative Algebra. Cambridge University Press, 1996.

- A. VERA. Introducción al Álgebra. (2 volúmenes). AVL, 1986.