Let $R$ be a commutative ring with $1 \ne 0$.
I'm trying to prove that if $R$ contains an ideal $I$ that is not finitely generated, then $R$ contains a proper ideal $J$ such that $R/J$ is Noetherian.
If $J$ is a maximal ideal, then $R/J$ would be a field, which would be Noetherian. Locating a maximal ideal is proving difficult for me, however. I know that since $I$ is not finitely generated, $R$ is not Noetherian. Thus I can't guarantee myself a maximal element under inclusion for a given nonempty set of ideals. I would appreciate some help finding a suitable maximal ideal, or a nudge in another direction if you know one.