Matej Bresar - Introduction to Noncommutative Algebra

Springer International Publishing Switzerland 2014

Matej Brear 1, 2

Finite dimensional division algebras are as simple and spotless as a (not necessarily commutative) ring can be. Their definition gathers together all of the most favorable properties. They are also the oldest, the most classical topic of noncommutative algebra. But as it often happens, the simplest objects are not the easiest to study, and classical themes are not less profound than modern ones. In many algebra textbooks finite dimensional division algebras are treated in the last chapters by using advanced tools. Our goals in this first chapter, however, are relatively modest. We will give an elementary, self-contained introduction to finite dimensional algebras, which in particular includes two results of extreme importance and beauty: Frobenius theorem on real division algebras and Wedderburns theorem on finite division rings. We shall not strictly confine ourselves to division algebras; presenting some of the results in the context of (central) simple algebras, even not necessarily finite dimensional ones, will not change the level of complexity of arguments and shall prove useful later.