simple module

Definition

Let be an R-module. is called simple or irreducible if it has no non-zero, proper submodules.

Relation to cyclic modules

Every simple module is cyclic.

Proof

Let be simple. Let generate . Then, for each , we have which is a submodule of , therefore by assumption or . If then it doesn’t contribute anything, and we don’t need it in the generating set, and we can consider . If then M is cyclic.

Converse is not true

Note, not every cyclic module is simple. Consider the -module . , but is a submodule, so is cyclic but not simple.

Examples

1. -modules can just be thought of as abelian groups, so the the simple modules are the groups of prime order.

2. -modules are just vector spaces, so the only simple modules are the 1 dimensional vector spaces, which are all isomorphic to .

3. The case is a lot more interesting.

   is a PID. Therefore, from the classification of finitely generated modules of PID modules for any -module (let )

   If is simple then it must be cyclic, so the only option are or for some prime . We know that as a module has proper submodules (since these the ideals of ) so we are left with . In order for this to be simple must be an irreducible polynomial, and since is algebraically closed is of the form . Therefore, the simple modules are