Hilbert's basis theorem tells us that $R[x]$ is Noetherian if $R$ is Noetherian. So, if every ideal $I\subset R$ is finitely generated, then every ideal $J\subset R[x]$ is finitely generated. This led me to consider the following:
Claim For any ring $R$ and ideal $I\subset R[x]$, if $I\cap R$ is finitely generated as an ideal of $R$, then $I$ is finitely generated as an ideal of $R[x]$.
I suspect this is in fact false, because if it were true it would make the proof of Hilbert's basis theorem 'too easy'. So, I'm trying to come up with a counter-example. So I need a (non-Noetherian) ring $R$, an ideal $I\subset R[x]$ that is not finitely generated, with $I\cap R$ finitely generated as an ideal of $R$.
Any help in finding a counter-example is appreciated. So far I've been considering ideals of $R[x_1]=\mathbb{C}[x_1,x_2,x_3,...]$ whose intersection with $R=\mathbb{C}[x_2,x_3,...]$ are finitely generated, and haven't got a counter-example yet.
You could take an ideal $J$ of $R$ that is not finitely generated, and let $I=xJ[x]$.
Then $I$ is not finitely generated as an ideal of $R[x]$ (if it were, then the coefficients of $x$ in a finite set of generators would generate $J$ as an ideal of $R$), but $I\cap R=0$ is finitely generated as an ideal of $R$.