Question: Let $R$ be a Noetherian ring, and $M$ be an $R-$module, show that $M$ is Noetherian if and only if $M$ is finitely generated.
This is a question on my homework, I'm really confused about one thing we proved in class the following theorem.
Theorem: Let $R$ be a ring and $M$ be any $R-$module. Then $M$ is Noetherian if and only if every submodule of $M$ is finitely generated.
Can't I just consider $M$ to be a submodule of it self to prove the question? Thank you.