Struktura končno razsežnih algeber

Strmčnik, Jaka (author), Brešar, Matej (mentor)

Besedilo obravnava osnovne vrste kolobarjev, kot so na primer enostavni kolobarji, prakolobarji, polprakolobarji, itd. Sproti se rezultate analogno prilagaja za algebre, saj je glavni cilj besedila dokazati Wedderburnov strukturni izrek za končno razsežne algebre. Precejšen poudarek je na pojmu modula. Potreben je tako za dokaze pomembnih rezultatov, kot tudi za vpeljavo pojma primitivnega kolobarja, ki je ključen pri formulaciji Jacobsonovega izreka o gostoti, na katerem sloni dokaz Wedderburnovega strukturnega izreka.

• Language: Slovenian
• Keywords: algebra, anihilator modula, gost kolobar linearnih operatorjev vektorskega prostora, nasprotni kolobar, nilpotenten ideal, $K$-modul, polprakolobar, prakolobar, primitiven kolobar, vektorski prostor nad obsegom
• Work type: Bachelor thesis/paper (mb11)
• Organization: FMF - Faculty of Mathematics and Physics
• Year: 2020
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Secondary language

• Language: English
• Title: Structure of Finite Dimensional Algebras

Abstract

This work deals with basic types of rings, such as simple rings, prime and semiprime rings, etc. Along the way, the results are being adjusted to hold also for algebras, since the main goal of the thesis is to prove Wedderburn’s structure theorem for finite dimensional algebras. The concept of a module is emphasised in the text. It is often used when proving some of the important results, and, moreover it plays a key role in introducing the notion of a primitive ring. The latter is used in the formulation of Jacobson Density Theorem, which provides the neccessary tool to prove Wedderburn’s structure theorem.

• Keywords: algebra, annihilator of a module, dense ring of linear operators of a vector space, ideal, opposite ring, nilpotent ideal, $K$-module, semiprime ring, prime ring, primitive ring, vector space over a division ring