A semisimple module is, informally, a module that is not far removed from simple modules. Specifically, it is a module with the following property: for every submodule , there exists a submodule such that and , where by 0 we mean the zero module.
Classification of semisimple modules
It happens that semisimple modules have a convenient classification (assuming the axiom of choice). To prove this classification, we first state some intermediate results.
Proof. First, suppose that is a submodule of . Let be a submodule of , and let be a submodule of such that and . We note that if and are elements such that , then . It follows that Since , it follows that is semisimple.
Now let us consider a quotient module of , with the canonical homomorphism . Let be a submodule of . Then is a submodule of , so there exists a submodule such that and . Then in , since is surjective, Therefore is semisimple as well.
Lemma 1. Let be a ring, and let be a nonzero cyclic left -module. Then contains a maximal proper submodule.
Proof. Let be a generator of . Let be the set of submodules that avoid , ordered by inclusion. Then is nonempty, as . Also, if is a nonempty chain in , then is an element of , as this is a submodule of that does not contain . Then is an upper bound on the chain ; thus every chain has an upper bound. Then by Zorn's Lemma, has a maximal element.
Lemma 2. Every cyclic semisimple module has a simple submodule.
Proof. Let be a cyclic semisimple module, and let be a generator for . Let be a maximal proper submodule of (as given in Lemma 1), and let be a submodule such that and . We claim that is simple.
Indeed, suppose that is a nonzero submodule of . Since the sum is direct, it follows that the sum is direct. Since strictly contains , it follows that , so ; it follows that ; thus is simple.
Theorem. Let be a left -module, for a ring . The following are equivalent:
- is a semisimple -module;
- is isomorphic to a direct sum of simple left -modules;
- is isomorphic to an (internal) sum of -modules.