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:
M5110/01: Thu 15:00–15:50 MS1,01016, L. Vokřínek
Prerequisites
M2110 Linear Algebra and Geometry 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
Abstract
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.
Key topics
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
Study resources and 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
Approaches, practices, and methods used in teaching
The course is offered two hours each week plus one hour of exercises. It initiates a discussion with students.
Method of verifying learning outcomes and course completion requirements
The course is ended by an oral exam.
Language of instruction
Czech
Teacher's information
Požadavky na úspěšné ukončení předmětu: 1. Porozumění základním pojmům (moduly, homomorfismy, podmoduly, faktorové moduly, součiny, přímé součty, tenzorové součiny 2. Znalost základů teorie projektivních, plochých a injektivních modulů 3. Pochopení návaznosti teorie modulů na lineární algebru a souvislosti s univerzální algebrou.
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 2011, Autumn 2013, Autumn 2015, autumn 2017.
  • Enrolment Statistics (Autumn 2009, recent)
  • Permalink: https://is.muni.cz/course/sci/autumn2009/M5110