PřF:M5110 Rings and Modules - Course Information
M5110 Rings and Modules
Faculty of ScienceAutumn 2022
The course is not taught in Autumn 2022
- Extent and Intensity
- 2/1. 3 credit(s) (příf plus uk k 1 zk 2 plus 1 > 4). Type of Completion: zk (examination).
- Teacher(s)
- prof. RNDr. Jiří Rosický, DrSc. (lecturer)
Mgr. Ivan Di Liberti, Ph.D. (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 - Prerequisites
- M2110 Linear Algebra II || ( FI:MA004 Linear Algebra and Geometry II )
Algebra: vector spaces, rings - Course Enrolment Limitations
- The course is also offered to the students of the fields other than those the course is directly associated with.
- fields of study / plans the course is directly associated with
- Algebra and Discrete Mathematics (programme PřF, N-MA)
- Applied Informatics (programme FI, N-AP)
- Geometry (programme PřF, N-MA)
- Course objectives
- The course introduces students to the theory of modules, one of fundamental topics of modern algebra. In particular:
*explains basic notions (modules, homomorphisms, submodules, quotient modules, products, direct sums, tensor products);
*presents the basics of the theory of projective, flat and injective modules and their structure properties;
*emphasizes an understanding of module theory as an extension of linear algebra and connections to universal algebra. - Learning outcomes
- Ability to
*explain basic notions (modules, homomorphisms, submodules, quotient modules, products, direct sums, tensor products);
*know the basics of the theory of projective, flat and injective modules and their structure properties;
*understand module theory as an extension of linear algebra and connections to universal algebra;
apply module theory in other fields of mathematics. - Syllabus
- Modules: modules, submodules, homomorphisms, quotient modules, products, direct sums, kernels, cokernels 2. Free and projective modules: free modules, projective modules, semisimple rings, vector spaces 3. Tensor product: tensor product and its properties 4. Flat modules: flat modules, directed colimits, Lazard's theorem, regular rings 5. Short exact sequences: short exact sequences, group Ext 6. Injective modules: injective modules, injective hull
- Literature
- L.Rowen, Ring theory I, Academic Press 1988
- A.J.Berrick, M.E.Keating, An introduction to rings and modules, Cambridge Univ. Press 2000
- Teaching methods
- The course is offered two hours each week plus one hour of exercises. It initiates a discussion with students.
- Assessment methods
- Course ends by an oral exam. Presence at the course is recommended, at the exercises is obligatory. Homeworks are given but not controled.
- Language of instruction
- Czech
- Further comments (probably available only in Czech)
- Course is no more offered.
The course is taught: every week.
- Enrolment Statistics (Autumn 2022, recent)
- Permalink: https://is.muni.cz/course/sci/autumn2022/M5110