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I've been thinking about the following Rotman's exercise, and just can't find an answer:

Give an example of a commutative ring $R$ containing proper ideals $I\subsetneq J\subsetneq R$ with $J$ finitely generated but with $I$ not finitely generated.

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Recall that if $R$ is Noetherian then any ideal is finitely generated. Hence to write down a counterexample you need a ring that is not Noetherian. For my money, the simplest example of such a ring is a polynomial algebra $k[x_1, x_2, \dots]$ in countably many variables. $J = (x_1)$ is a finitely generated proper ideal, and it has a proper subideal

$$I = (x_1 x_2, x_1 x_3, x_1 x_4, \dots)$$

which I claim is not finitely generated. Do you see why?

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    $\begingroup$ Nice one @QiaochuYuan $\endgroup$ – layman Mar 25 '15 at 21:41

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