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I'm looking for a a finitely generated module with submodules that are not finitely generated. I know about $\Bbb R$ and $\Bbb Z$, but I need another example.

Could anyone help me? I'm new in algebra.

Xam
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Math90
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1 Answers1

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You need to consider a non-Noetherian ring. Such a ring will have an ideal that is not finitely generated. For an example take $R=\Bbb Z[X_1,X_2,\ldots]$ a polynomial ring in infinitely many variables. The ideal generated by all the $X_j$ is not finitely generated.

Over a Noetherian ring, all submodules of finitely generated modules are finitely generated.

Angina Seng
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