Here is the question:
Let $R$ be a countable Noetherian ring, and let $C$ be the equivalence classes of left cyclic $R$-modules. Show that $C$ is countable.
I started with countable fields (or division algebras): any cyclic $R$-modules will be a $1$-dimensional vector space over $R$, so $|C| = 1$, hence at most countable.
More generally, any cyclic $R$-module is a finitely generated module over Noetherian $R$, thus a Noetherian $R$-module. Then any ascending chain inside each cyclic $R$-module has finite length and every submodule in every chain is finitely generated.
Hence from here I conclude that $|C| \le (\# \text{chains})(\text{max chain length})$. Since $\# \text{chains}$ is countable (every element in a chain has to be finitely generated, and thus there are at most countable generators for things in a chain), this bounds $C$ to be a countable set.
I'm NOT sure if my argument is correct and it also seems a bit lengthy. Could someone see it more directly? Any help is appreciated!