Let there be an exact sequence of $R$-modules: $$0\rightarrow A\rightarrow B\rightarrow C\rightarrow 0$$ where $R$ is a principal ideal domain and $B$ is finitely generated.
Are $A$ and $C$ finitely generated?
I know they are, because $C$ is isomorphic with $B/A$ and $A$ is isomorphic with a submodule of $B$ and PIDs are noetherian. But I don't fully understand this reasoning, could someone explain it a bit more?