Bergman's centralizer theorem : master's thesis

Snoj, Jakob Jurij (author), Smertnig, Daniel (mentor)

Bergman's centralizer theorem is a theorem describing the structure of a centralizer of a non-scalar element in a free associative algebra over a field. The thesis aims to present a reasonably self-contained proof of this result, following both the original proof by G. M. Bergman and a slightly different approach by P. M. Cohn, highlighting the main differences between the two approaches. A large part of the thesis is dedicated to developing tools from the theory of free ideal rings (firs) which are a main component of the proof. The proof mainly uses methods from noncommutative algebra, but tools from other areas are also introduced, such as valuation theory. In the final chapter, we demonstrate a new result generalizing a step in Bergman's proof of the centralizer theorem to a different structure.

• Language: English
• Keywords: Bergman's centralizer theorem, free associative algebra, centralizer
• Work type: Master's thesis/paper
• Typology: 2.09 - Master's Thesis
• Organization: FMF - Faculty of Mathematics and Physics
• Year: 2025
• PID: 20.500.12556/RUL-171607
• UDC: 512
• COBISS.SI-ID: 246679299
• Publication date in RUL: 29.08.2025

Secondary language

• Language: Slovenian
• Title: Bergmanov izrek o centralizatorju

Abstract

Bergmanov izrek o centralizatorju je izrek, ki opisuje strukturo centralizatorja neskalarnega elementa v prosti asociativni algebri nad poljem. Cilj magistrskega dela je predstaviti dokaz tega rezultata, ki sledi tako prvotnemu dokazu G. M. Bergmana kot tudi nekoliko drugačnem pristopu P. M. Cohna, ter poudariti razlike med omenjenima dokazoma. Velik delež magistrskega dela je posvečen obravnavi teorije kolobarjev s prostimi ideali (fir-ov), ki tvorijo eno izmed glavnih komponent dokaza. Dokaz večinoma uporablja orodja nekomutativne algebra, predstavimo pa tudi nekaj orodij iz drugih področij, kot denimo teorije valuacijskih kolobarjev. V zadnjem poglavju predstavimo nov rezultat, ki posploši enega izmed korakov v Bergmanovem dokazu izreka o centralizatorju na nekoliko drugačno strukturo.

• Keywords: Bergmanov izrek o centralizatorju, prosta asociativna algebra, centralizator