Example of a finitely generated ideal $I$ with $R/I$ is Noetherian and $R$ not Noetherian.

Question

Can somebody give me example of ring $R$ such that $R/I$ is Noetherian but $R$ is not Noetherian ring? $I$ is finitely generated ideal of $R$.

Also please search example such that $I$ is not nilpotent because if $I$ is nilpotent then R become Notherian ring.

Answer 1

$R=\prod_{i\in \mathbb N}F_2$, with $I=(0,1,1,\ldots)R$.

$R/I$ is the field $F_2$ and $I$ is principal.