Wedderburn Theory

At the center of this chapter stands the Wedderburn structure theorem, according to which every simple artinian algebra is isomorphic to a matrix algebra Mn(D) over some division algebra D, with n and (the isomorphism class of) D uniquely determined. A structure result in abstract algebra, and a very satisfying one at that, which one can prove through simple methods of linear algebra! (This was first done by E. Artin.) It reduces the study of simple artinian algebras to that of division algebras and thus represents not only an achievement but also a starting point for further investigations, in that it leads us to pursue a classification of division algebras. This problem turns out to be tougher than it may appear at first, even after making further restrictions; nonetheless we will be able to deal in Chapter 31 with the case of local division algebras.