PřF:M7350 Algebra III - Course Information
M7350 Algebra III
Faculty of ScienceAutumn 2024
- Extent and Intensity
- 2/1/0. 5 credit(s). Recommended Type of Completion: zk (examination). Other types of completion: k (colloquium).
In-person direct teaching - Teacher(s)
- prof. RNDr. Jiří Rosický, DrSc. (lecturer)
Kristóf Kanalas, MSc (seminar tutor) - Guaranteed by
- prof. RNDr. Jiří Rosický, DrSc.
Department of Mathematics and Statistics – Departments – Faculty of Science
Supplier department: Department of Mathematics and Statistics – Departments – Faculty of Science - Timetable
- Mon 10:00–11:50 M3,01023
- Timetable of Seminar Groups:
- Prerequisites
- Algebra I, Algebra II
- Course Enrolment Limitations
- The course is offered to students of any study field.
- Course objectives
- The goal is to continue the bachelor's course of algebra and acquaint students with the choosen areas of modern algebra. In particular:
- to introduce the language of category theory and illustrate it on examples;
- to explain the concept of an universal algebra with the emphasis on terms and free algebras;
- to present the basics of the theory of mudules with the emphasize on free, projective, flat and injective modules. - Learning outcomes
- After the course, students should be able:
- to think in the language of category theory;
- to apply the basic ideas of universal algebra, including terms and free algebras;
- to understand module theory as the generalization of linear algebra;
- to apply the acquired knowledge to other areas of mathematics. - Syllabus
- 1. Categories: categories, functors, natural transformations, examples. 2. Universal algebras: universal algebras, subalgebras, products, factor algebras, terms, free algebras, Birkhoff's theorem. 3. Modules: modules, submodules, homomorphisms, factor modules, products, direct sums, kernels, cokernels. 4. Free and projective modules: free modules, projective modules, semisimple modules. 5. Tensor product: tensor product and its properties. 6. Flat modules: flat modules, directed colimits, Lazard's theorem, regular rings. 7. Injective modules: injective modules, injective hull.
- Literature
- Jiří ROSICKÝ, Algebra 3, učební text
- J. R. Rotman, Advanced Modern Algebra, AMS 2017
- BICAN, Ladislav and Jiří ROSICKÝ. Teorie svazů a univerzální algebra. Vyd. 1. Praha: Ministerstvo školství, mládeže a tělovýchovy ČSR, 1989, 84 s. info
- Teaching methods
- Lectures: theoretical explanation based on given literature suplemented with examples and applications. It will be in presence or, in the case of need, on-line.
Exercises: solving theoretical problems focused on practising basic concepts and theorems - Assessment methods
- Course ends by an oral exam. Exams will be in presence or, in the case of need, online using Zoom. Presence at the course recommended. Homeworks are given, handed in exercises.
- Language of instruction
- English
- Further Comments
- Study Materials
The course is taught annually.
- Enrolment Statistics (recent)
- Permalink: https://is.muni.cz/course/sci/autumn2024/M7350