#
PřF:M5110 Rings and Modules - Course Information

## M5110 Rings and Modules

**Faculty of Science**

Autumn 2015

**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)

John Denis Bourke (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**- Tue 14:00–15:50 M6,01011
- Timetable of Seminar Groups:

*J. Bourke* **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.

At the end of the course students should be able 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 geometry and topology. **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**- A.J.Berrick, M.E.Keating, An introduction to rings and modules, Cambridge Univ. Press 2000
- L.Rowen, Ring theory I, Academic Press 1988

**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)**- The course is taught once in two years.

- Enrolment Statistics (Autumn 2015, recent)
- Permalink: https://is.muni.cz/course/sci/autumn2015/M5110