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PřF:M5110 Rings and Modules - Course Information

## M5110 Rings and Modules

**Faculty of Science**

Autumn 2009

**Extent and Intensity**- 2/1. 3 credit(s) (fasci plus compl plus > 4). Type of Completion: zk (examination).
**Teacher(s)**- prof. RNDr. Jiří Rosický, DrSc. (lecturer)

doc. Lukáš Vokřínek, PhD. (lecturer) **Guaranteed by**- prof. RNDr. Jiří Rosický, DrSc.

Department of Mathematics and Statistics - Departments - Faculty of Science **Timetable**- Thu 10:00–11:50 M2,01021
- Timetable of Seminar Groups:

*L. Vokřínek* **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**- Applied Informatics (programme FI, N-AP)
- Informatics (programme FI, N-IN)
- Mathematics (programme PřF, M-MA)

**Course objectives**- The course presents one of fundamental topics of modern algebra. It naturally follows the well-known theory of vector spaces and shows what happens if scalars form a ring and not a field. It presents the emerging concepts of projective, flat and injective modules and their structure properties. Doing it, there are used basic constructions like products, direct sums, kernels, cokernels and tensor products. The course prepares students to the use of modules 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 sequaences: 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**- The course is ended by an oral exam.
**Language of instruction**- Czech
**Further comments (probably available only in Czech)**- The course is taught once in two years.

- Enrolment Statistics (Autumn 2009, recent)
- Permalink: https://is.muni.cz/course/sci/autumn2009/M5110