Every (non-zero) finitely generated module over a ring has a maximal proper submodule by a simple application of Zorn's lemma.
I am interested in the following question, with two variants.
Variant 1: For which groups $G$ does every (non-zero) $\mathbb CG$-module have a maximal proper submodule.
Variant 2: For which groups $G$ does every countably generated (non-zero) $\mathbb CG$-module have a maximal proper submodule.
What I know about this:
- If $G$ is finite, then the answer to Variant 1 (and hence 2) is yes since every (non-zero) module $M$ over an Artinian ring $R$ has a maximal proper submodule (if $J$ is the radical of $R$, then $JM$ is proper because $J$ is nilpotent and so $M/JM$ is semisimple and hence has a simple quotient, whose kernel pulls back to a maximal submodule of $M$)?
- If $G$ is commutative, then by the first theorem of this paper Variant 1 holds for $\mathbb CG$ iff $G$ is locally finite (using complex group rings are semiprimitive and are von Neumann regular iff the group is locally finite)?