Estrutura de álgebras associativas com dimensão finita

Abstract

The present work aims to introduce concepts and properties of finite-dimensional associative algebras, in order to describe their structure. In the first chapter, we will present properties of a few algebraic structures, starting with the proof of Frobenius theorem. After that, we will define simple rings, central algebras, multiplication algebras, automorphisms of simple central algebras and maximal subfields, in order to finish with the proofs of the Skolem-Noether and Wedderburn theorems. In the second chapter, we will focus on Wedderburn’s structure theory. We will define the concepts of nilpotent ideals, prime and semiprime rings, unitization of algebras, regular representation, group algebras, matrix units, idempotents and minimal ideals, in order to prove Maschke’s theorem and Wedderburn’s structure theorems, which concretely describe the structure of finite-dimensional associative algebras. This work will conclude with some practical examples that illustrate these results.