## 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:
M5110/01: Tue 16:00–16:50 M6,01011, 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
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.
The course is also listed under the following terms Autumn 2007 - for the purpose of the accreditation, Autumn 1999, Autumn 2000, Autumn 2001, Autumn 2003, Autumn 2005, Autumn 2007, Autumn 2009, Autumn 2011, Autumn 2013, autumn 2017.
• Enrolment Statistics (Autumn 2015, recent)