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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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A submodule of a finitely generated module over a noetherian ring is finitely generated. Find a non-noetherian ring. The regular module is finitely generated, but by definition it has submodules that are not. –  Jack Schmidt Nov 17 '11 at 15:12
@JackSchmidt: I'm sorry that I do not know enough examples for such a ring. Could you please give a specific example? Thank you! –  Roun Nov 17 '11 at 15:28

2 Answers 2

up vote 11 down vote accepted

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 Georges. Can one argue as follows? Assume $I$ was finitely generated. Then $I = \langle X_{n_1}, \dots , X_{n_N} \rangle$. Without loss of generality, we may assume $n_1 < \dots < n_N$. Then the polynomial $p(X_1, X_2, \dots) = X_{n_N + 1}$ is not in $I$. Hence $I$ cannot be finitely generated. –  Rudy the Reindeer Jul 23 '12 at 8:51
Dear Matt, beware that if $I$ were finitely generated there is no reason that it would be generated by monomials $X_{n_i}$ ! –  Georges Elencwajg Jul 23 '12 at 9:19

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