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.