MODULE THEORY
Academic year and teacher
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- Versione italiana
- Academic year
- 2021/2022
- Teacher
- CLAUDIA MENINI
- Credits
- 6
- Didactic period
- Secondo Semestre
- SSD
- MAT/02
Training objectives
- The aim of the course is to provide some basic concepts in Ring and Module Theory together with an introduction to Category Theory.
In particular to let the students know about the construction of sums, products and tensor products of modules, and the characterization of projective and injective modules, and moreover to get acquainted with abelian and Grothendieck categories.
And the end of the course the student will be able to prove existence and properties of the studied constructions and will be also able to recognise them and use them in order to study other mathematical objects. Prerequisites
- The usual program of the Algebra course. In particular familiarity with the concept of algebraic structure, of morphism between structures and the basic properties of groups, rings and vector spaces.
Course programme
- Modules and their morphisms. Quotient theorem and isomorphism theorems (4 hours).
Direct product for a family of left R-modules and direct sum of a family of left R-module (6 hours).
Short exact sequences and split exact sequences (4 hours).
Free and Projective Modules (3 hours).
Injective and divisible modules. Baer criterion for injectivity. Torsion Groups (4 hours).
Generators and cogenerators. Simple left modules and maximal left ideals (4 hours).
Tensor product of modules, flat Modules, bimodules, adjunction between tensor product and Hom functor (12 hours).
Categories, functors and natural transformations. Equivalence and isomorphism of categories (5 hours).
Abelian and Grothendieck categories (5 hours). Didactic methods
- Lectures
Learning assessment procedures
- Oral examination. The candidate will present a subject either taken from those developed during the course or of her/his own choice. In the latter case the subject will be decided together with the teacher.
During the presentation the candidate is supposed to show his/her acquisition of basic contents and techniques of module theory.
In order to show a level of excellence the candidate is supposed to be able to solve problems relative to module theory not previously discussed during classes. Reference texts
- C. Menini "Module Theory" Lecture Notes 2014/15
C. Menini "Category Theory" Lecture Notes 2014/15
F. Anderson, K. R. Fuller, Rings and categories of modules. Second edition. Graduate Texts in Mathematics, 13. Springer-Verlag, New York, 1992.
