Modules that have only finitely many submodules

Question

Drawing the lattice of submodules of a given module helps me to gain some intuition about the structure of module. Sometimes, however, it is not possible to draw in neat manner; For example vector spaces may have infinitely many subspaces. What is a good class of modules to draw the lattice of submodules in neat manner? (Uniserial modules are candidate which I know of.) An answer to this question might be somewhere, but I don't know. Lastly, let me make my questions clear.

1. What is a (large enough) class of modules that have only finitely many submodules (NOT up to isomorphic)?
2. What is a (large enough) class of rings or algebras that every (finitely generated) right modules have only finitely many submodules? Solved owing to Jeremy Rickard's answer.

Answer 1

It seems like your best bet for both questions will be to consider finite rings and their finitely generated modules. These at least will be closed under products.

Answer 2

For the second question, the finite rings are precisely those for which every finitely generated module has finitely many submodules.

For a ring $R$ and $r\in R$, let $M_r$ be the submodule of $R\oplus R$ generated by $(1,r)$. Then $(1,r)$ is the only element of $M_r$ whose first coordinate is $1$, and so $M_r\neq M_s$ for $r\neq s$, and so if $R$ is infinite then the modules $\{M_r\mid r\in R\}$ form an infinite set of submodules of $R\oplus R$.

Answer 3

There is a paper on modules with finitely many submodus, by Akbari, Khalashi Ghezelahmad and Yaraneri.