Semisimple Modules & Rings and the Wedderburn Structure Theorem

Abstract

This chapter is concerned with looking at part of a structure theory for rings. The idea of any “structure theory” of an object (in this case a ring) is to express that object in terms of simpler, better understood pieces. For example, the Wedderburn Structure Theorem says that any semisimple ring (we’ll define this later) is isomorphic to a finite product of matrix rings over division rings, each of which is simple. The theory for semisimple modules is in many ways analogous to the theory of vector spaces over a field, where we can break up vector spaces as sums of certain subspaces.