Wedderburn-Artin Theory

Abstract

Modern ring theory began when J.H.M. Wedderburn proved his celebrated classification theorem for finite dimensional semisimple algebras over fields. Twenty years later, E. Noether and E. Artin introduced the Ascending Chain Condition (ACC) and the Descending Chain Condition (DCC) as substitutes for finite dimensionality, and Artin proved the analogue of Wedderburn’s Theorem for general semisimple rings. The Wedderburn-Artin theory has since become the cornerstone of noncommutative ring theory, so in this first chapter of our book, it is only fitting that we devote ourselves to an exposition of this basic theory.