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I'm looking for an example of a finitely generated module with submodules that are not finitely generated.

I've found a similar question dealing with group (i.e. an example of a finitely generated group with subgroups that are not finitely generated). But I can't figure out whether that question do help to this one.

And I actually want to find a more "module-like" example rather than an example derived from a 'strange' group.

Can you please help? Thank you!

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    Plus one because over a month ago I proved that every quotient of a fin gen module is fin gen and then made the mistake of assuming sub mods were too.2017-03-19

3 Answers 3

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Here's a fairly simple example (of a non-Noetherian ring): the ring $R$ of polynomials in one indeterminate $X$ having rational coefficients but with an integer constant term. Its ideal of elements with zero constant term is not finitely generated as an $R$-module.

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The ideal $I=\langle X_1,X_2,...,X_n,... \rangle \subset \mathbb R[X_1,X_2,...,X_n,...]=A$ can be seen as a submodule of the free $A$-module of dimension one $A=A^1$, and that module is not finitely generated. Do you see why?
(Hint: even in a polynomial ring with infinitely many indeterminates, each polynomial involves only finitely many variables. In other words $\mathbb R[X_1,X_2,...,X_n,...]=\bigcup_{k\geq 1}\mathbb R[X_1,X_2,...,X_k] \;$ )

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    Dear Matt, beware that if $I$ were finitely generated there is no reason that it would be generated by monomials $X_{n_i}$ !2012-07-23