Definition of finitely generated submodule

Question

I have a slight issue with the definition of when a submodule is finitely generated. Let $R$ be a ring, $M$ be an $R$-module and $N$ be a submodule of $M$. Which of these is correct?

(1) $N$ is finitely generated if it is generated by finite $A\subseteq M$.

(2) $N$ is finitely generated if it is generated by finite $A\subseteq N$.

Answer 1

If (2) holds, then $A\subseteq N\subseteq M$, so (1) holds. If (1) holds, then it must be the case that $A\subseteq N$, or else $N<\langle A\rangle$, so (2) holds.