Structure of rings and applications

• Grant number: 23/13047-7
• Support Opportunities: Scholarships in Brazil - Scientific Initiation
• Start date: December 01, 2023
• End date: November 30, 2024
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Algebra
• Principal Investigator: Plamen Emilov Kochloukov
• Grantee: Gabriel Cruz Vitale Torkomian
• Host Institution: Instituto de Matemática, Estatística e Computação Científica (IMECC). Universidade Estadual de Campinas (UNICAMP). Campinas , SP, Brazil
• Associated research grant: 18/23690-6 - Structures, representations, and applications of algebraic systems, AP.TEM

Abstract

In this research project we shall study the structure of rings and algebras, towards some applications. We shall start with the basic properties of the rings such as primitiveness, semi-simplicity, the Jacobson radical, the Chain conditions, then we shall go to simple rings, and the Wedderburn--Artin theorem. We shall go into details about simple algebras and the Skolem--Noether theorem, and also the Double centralizer theorem. As an application we shall revisit the classical representation theory of finite groups over the complex numbers from the point of view of the structure of rings and algebras. Another application will be concerned with matrix groups and the classical finiteness Burnside theorems. Evidently we shall also see the Golod--Shafarevich construction. The project will be based on the monographs by M. Bresar, Introduction to noncommutative algebra, and by I. Herstein, Noncommutative rings.