Can someone give an example of a ring $R$, a left $R$-module $M$ and a submodule $N$ of $M$ such that $M$ is finitely generated, but $N$ is not finitely generated?

I tried a couple of examples of modules I know and got nothing...


marked as duplicate by user26857, Jyrki Lahtonen abstract-algebra Jun 7 '16 at 20:31

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Consider the simplest possible nontrivial (left) $R$-module: $R$ itself. It's certainly finitely generated, by $\{ 1 \}$. The submodules are exactly the (left) ideals of $R$. So you want a ring which has (left) ideals which are not finitely generated. For example, you could use a non-Noetherian commutative ring, such as $\mathbb{Z}[X_1, X_2, X_3, \ldots ]$.

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    $\begingroup$ Even more is true: A ring $R$ is noetherian if and only if submodules of f.g. modules over $R$ are f.g. $\endgroup$ – Martin Brandenburg Mar 27 '12 at 10:46

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