M7350 Algebra III

Faculty of Science
Autumn 2025
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)
doc. John Denis Bourke, PhD (lecturer)
prof. RNDr. Jiří Rosický, DrSc. (lecturer)
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 8:00–9:50 M6,01011
  • Timetable of Seminar Groups:
M7350/01: Thu 16:00–16:50 M1,01017, J. Bourke
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
There will be 2-3 marked assignments, worth 20 percent in total of the final grade. Examination will be by a 2 hour written exam.
Language of instruction
English
Further Comments
Study Materials
The course is taught annually.
The course is also listed under the following terms Autumn 2019, Autumn 2020, autumn 2021, Autumn 2022, Autumn 2023, Autumn 2024.
  • Enrolment Statistics (recent)
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