Associative algebras

• Grant number: 26/14795-5
• Support Opportunities: Scholarships in Brazil - Scientific Initiation
• Start date: September 01, 2026
• End date: August 31, 2027
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Algebra
• Principal Investigator: Plamen Emilov Kochloukov
• Grantee: Mateus José da Silva Costa
• Host Institution: Instituto de Matemática, Estatística e Computação Científica (IMECC). Universidade Estadual de Campinas (UNICAMP). Campinas , SP, Brazil
• Associated research grant: 24/14914-9 - Structures and representations of algebraic systems and their applications, AP.TEM

Abstract

This undergraduate research project will address the basics of the theory of rings and associative algebras. The student is pursuing a bachelor's degree in mathematics and has already completed advanced courses: advanced linear algebra, number theory, and is currently taking a course on groups and representations. He has already completed an undergraduate research project in Algebra (with another supervisor), funded by this Foundation, and therefore we consider that he has the prerequisites to begin studies on this project. We will begin our studies with division algebras, and we will prove Frobenius' theorem on division algebras over the real numbers. We will study the classical theory of finite-dimensional algebras. We will explore extensions of various notions from linear algebra: modules, semisimplicity, and Schur's lemma. As an extension of linear algebra notions, we will examine the concepts of Artinian and Noetherian modules and rings. We will review the tensor product of modules and vector spaces, and then we will look at the tensor product of algebras. Applications will include Skolem-Noether's theorem, as well as the double centralizer theorem. This will naturally lead us to the Brauer group and its fundamental properties. We will see what the Brauer group of complex numbers and the Brauer group of real numbers are and then also show that the Brauer group of rational numbers is infinite. We will study groups of matrices and Burnside's theorem, and of course, the Golod-Shafarevich construction. We will conclude our studies with the notion of primitiveness, Jacobson's radical, and applications. The objective of this project is to introduce the student to the world of modern algebra, with abstract concepts and important theorems. Thus, this undergraduate research project can contribute to Mateus José's professional development, enrich his mathematical culture, and better prepare him for postgraduate studies in mathematics. (AU)