Classical Ring-Structure Theorems

Abstract

As we saw in the last chapter semisimple modules play a distinguished role in the theory of modules. Classically, the most important class of rings consists of those rings R whose category _R M has a semisimple generator. A characteristic property of such a ring R, called a “semisimple” ring, is that each left R-module is semisimple. These rings are the objects of study in Section 13 where we prove the fundamental Wedderburn-Artin characterization of these rings as direct sums of matrix rings over division rings. In particular, a semisimple ring is a direct sum of rings each having a simple faithful left module. In Section 14 we study rings characterized by this latter property—the “(left) primitive” rings. Here we prove Jacobson’s important generalization of the semisimple case characterizing left primitive rings as “dense rings” of linear transformations.