# Finitely generated ideal containing non finitely generated ideal

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.

• – layman Mar 25 '15 at 16:48
• Any example in a polynomial ring? – Crisp Mar 25 '15 at 18:59

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