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

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

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Very good question! – Leon Jul 7 '13 at 1:01
up vote 25 down vote accepted

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

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