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Course, academic year 2023/2024
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Rings and Modules - NMAG333
Title: Okruhy a moduly
Guaranteed by: Department of Algebra (32-KA)
Faculty: Faculty of Mathematics and Physics
Actual: from 2021
Semester: winter
E-Credits: 5
Hours per week, examination: winter s.:2/2, C+Ex [HT]
Capacity: unlimited
Min. number of students: unlimited
4EU+: no
Virtual mobility / capacity: no
State of the course: not taught
Language: Czech
Teaching methods: full-time
Teaching methods: full-time
Guarantor: prof. RNDr. Jan Trlifaj, CSc., DSc.
Class: M Bc. OM
M Bc. OM > Zaměření MSTR
M Bc. OM > Povinně volitelné
Classification: Mathematics > Algebra
Incompatibility : NMAG339
Interchangeability : NMAG339
Is incompatible with: NMAG339
Is interchangeable with: NALG028, NMAG339
In complex pre-requisite: NMAG349
Annotation -
Last update: G_M (15.05.2012)
Completely reducible, artinian, and noetherian rings and modules. Free, projective, and injective modules. The Krull-Remak-Schmidt Theorem. Introduction to the representation theory of finite dimensional algebras. A recommended course for specialization Mathematical Structures within General Mathematics.
Course completion requirements -
Last update: doc. Mgr. et Mgr. Jan Žemlička, Ph.D. (28.10.2019)

Active participance in practicals.

Literature -
Last update: G_M (24.04.2012)

F.W.Anderson, K.R.Fuller: Rings and Categories of Modules, 2nd ed.,Springer, New York, 1992.

D. S. Passman, A Course in Ring Theory, AMS, Providence, 2004.

I. Assem, D. Simson, A. Skowronski, Elements of the Representation Theory of Associative Algebras. Vol. 1, Cambridge Univ. Press, Cambridge, 2006.

Requirements to the exam -
Last update: prof. RNDr. Jan Trlifaj, CSc., DSc. (05.10.2017)

For the exam, knowledge of the topics presented in the course is required (structure of completely reducible rings and modules, Wedderburn-Artin Theorem, artinian and noetherian rings and modules, Hopkins Theorem, modules of finite length, Jordan-Hoelder Theorem, structure of free and projective modules, Kaplansky Theorems, injective modules and their structure over noetherian rings). The exam is oral, ,zapocet' is not required for the exam.

Syllabus -
Last update: G_M (24.04.2012)

Contents of the Lecture: 

Ring theory (Jacobson radical, structure of completely reducible modules and rings, Wedderburn-Artin Theorem. Artinian and noetherian rings and modules, Hopkins Theorem, Hilbert Basis Theorem.)

Module theory (Free and projective modules, Kaplansky theorems. Injective modules, The Baer Criterion, injetive hulls, structure of injective modules over noetherian rings, structure of divisible abelian groups, hereditary rings).

Supplementary topic: Envelopes and covers of modules. Projective and flat covers.

Example class: 

Examples. The Krull-Remak-Schmidt Theorem. Elements of the representation theory of finite dimensional algebras (path algebras of quivers, their Jacobson radical and heredity, linear representations of quivers as modules over path algebras).

Entry requirements -
Last update: G_M (24.04.2012)

Knowledge of basics of algebra at the level of Algebra I (NALG026).

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