7 Rings with Semisimple Generators.
It is now quite easy to use Morita to obtain the classical Wedderburn and Artin-Wedderburn charac- terizations of simple Artinian and semisimple rings. We begin by reminding you of a few elementary facts about semisimple modules.
Recall that a module RM is simple in case it is a non-zero module with no non-trivial submodules.
7.1. Lemma. [Schur’s Lemma] If M is a simple module and N is a module, then
1. Every non-zero homomorphism M → N is a monomorphism;
2. Every non-zero homomorphism N → M is an epimorphism;
3. If N is simple, then every non-zero homomorphism M → N is an isomorphism;
In particular, End(RM) is a division ring.
A module RM is semisimple if it is the sum of its simple submodules. Then an easy, but important, characterization of semisimple modules is given in [1] (see Theorem 9.6):
7.2. Proposition. For left R-module RM the following are equivalent:
(a) M is semisimple;
(b) M is a direct sum of simple submodules;
(c) Every submodule of M is a direct summand.
Let’s begin with a particularly nice example. Indeed, let D be a division ring. Then the regular module DD is a simple module, and hence, a simple progenerator. In fact, every nitely generated module over D is just a nite dimensional D-vector space, and hence, free. That is, the progenerators for D are simply the non-zero nite dimensional D-vector spaces, and so the rings equivalent to D are nothing more than the n n matrix rings over D. That leads us to investigate what can be said about an arbitrary ring that has a simple generator.
Well, suppose that R is a ring and that it has a simple generator RT . But then T generates RR, and so R is a sum of (by Lemma 7.1) simple left ideals all isomorphic to T . Since 1 ∈ R, this means that there is a nite set of these simple left ideals that sum to R. Thus, there is a minimal such set, say, I1,...,In. By minimality, these left ideals must be independent, and so this sum must be direct. That is,
R = I1 In 54 Section 7