Radikal kolobarja : delo diplomskega seminarja

Snoj, Jakob Jurij (author), Brešar, Matej (mentor)

V delu je predstavljen Jacobsonov radikal kolobarja skupaj s povezano teorijo in nekaterimi lastnosti. Z uvedbo kvazi-obrnljivosti je dokazana simetričnost definicije Jacobsonovega radikala. Opazujemo lastnosti na levo artinskih kolobarjih ter dokažemo Wedderburn-Artinov izrek, ki podaja klasifikacijo levo artinskih kolobarjev s trivialnim Jacobsonovim radikalom. Dokažemo tudi Jacobsonov izrek o gostoti ter podamo aplikacijo teorije Jacobsonovega radikala.

• Language: Slovenian
• Keywords: Jacobsonov radikal, artinski kolobarji, Wedderburn-Artinov izrek, Jacobsonov izrek o gostoti
• Work type: Bachelor thesis/paper
• Organization: FMF - Faculty of Mathematics and Physics
• Year: 2023
• PID: 20.500.12556/RUL-149823
• UDC: 512
• COBISS.SI-ID: 163966723
• Publication date in RUL: 10.09.2023

Secondary language

• Language: English
• Title: Radical of a Ring

Abstract

The Jacobson radical of a ring is presented along with the related theory and certain properties. Quasi-invertibility is introduced with the focus of proving the symmetric property of the definition of the Jacobson radical. The Jacobson radical is further discussed in the context of Artinian rings and the Wedderburn-Artin Theorem is proven, providing a classification of left Artinian rings with a trivial Jacobson radical. The Jacobson Density Theorem is proven and an application of the Jacobson radical theory is demonstrated.

• Keywords: The Jacobson radical, Artinian rings, Wedderburn-Artin Theorem, Jacobson Density Theorem