Question

finitely generated submodule

Answer 1

You mean other than directly from the definition by exhibiting a finite generating set?

A submodule is, in particular, a *module*. So any result that says "if X is true then M is a finitely generated module" could potentially be helpful.

The nicest situation is if your submodule is a submodule of a finitely generated module over a Noetherian ring. In general, though, a submodule of a finitely generated module need not be finitely generated. So you need both of those conditions.