20.500.12556/RUL-171607 Bergman's centralizer theorem master's thesis Bergmanov izrek o centralizatorju 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. 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. Bergman's centralizer theorem free associative algebra centralizer Bergmanov izrek o centralizatorju prosta asociativna algebra centralizator true false false Angleški jezik Slovenski jezik Magistrsko delo/naloga 2025-08-29 08:15:04 2025-08-29 08:15:09 2025-09-08 03:49:35 0000-00-00 00:00:00 2025 0 0 0000-00-00 NiDoloceno NiDoloceno NiDoloceno 0000-00-00 0000-00-00 0000-00-00 512 151980 246679299 18195.pdf 18195.pdf 1 FEF4CDA0B7476D936A1DAF1667F95D50 116a3bdf123f995a67e21575ec53632eac0539051c1b0bbfee3c766435ba0da2 62259e7d-849f-11f0-9328-0050569b8976 https://repozitorij.uni-lj.si/Dokument.php?lang=slv&id=214987 Fakulteta za matematiko in fiziko 0 0 0