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 $A\subseteq N$.

score 0 · Accepted Answer · answered Mar 16 '20 at 00:21

0

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.

answered Mar 16 '20 at 00:21

Silly me! I don't know how I didn't spot that. Thanks – user829347 Mar 16 '20 at 00:50
Sometimes it just takes someone else to point at the seemingly obvious :) – Mar 16 '20 at 00:58

Definition of finitely generated submodule

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