Matrice nad nekomutativnim prstenima

Tema ovog radu su matrice nad nekomutativnim prstenom. U prvom i drugom poglavlju pod nazivom Prsteni i Moduli uvodimo osnovne pojmove koji su nam potrebni u daljnjem dijelu rada. Pojam poluprostota, kojim se bavimo u trećem poglavlju, je najosnovniji termin koji se proteze kroz čitav diplomski rad. Prezentiramo tri definicije poluprostog modula i dokazujemo da su one ekvivalentne. Pomoću navedenih uvjeta možemo dokazati Jacobsonov teorem o gustoći te definirati pojmove poluprostog i prostog prstena. Uz Jacobsonov teorem u radu dokazujemo i druge važne teoreme, kao primjer Wedderburnov teorem koji nam daje prikaz prstena R kao D-endomorfizam nad modulom E, gdje sa D označavamo konačnodimenzionalnu algebru sa dijeljenjem nad poljem k.The theme of this paper are matrices over noncommutative ring. In the first and second chapter titled Rings and Modules we introduce the basic concepts that are required in the far part of the work. The concept semisimplicity, the main theme of the third chapter, is the most basic term that runs through this entire graduate paper. We present three definitions of semisimple modules and prove that they are equivalent. Besides Jacobson’s theorem we also prove some others very important theorems, for example Wedderbun’s theorem which gives us a representation of a ring R as the ring of D-endomorphisms of module E, where D is a finite-dimensional division algebra over a field k. Using the above conditions we can prove Jacobson’s density theorem and define the terms of semisimple and simple ring