Commutative ring with some special kind of localizations being Noetherian

Question

Let $R$ be a commutative ring with unity. If for every non invertible element $f\in R$, we have $R_f$ (the localization of $R$ at the multiplicative set $\{f^n:n\ge 0\}$ ) is a Noetherian ring, then is $R$ a Noetherian ring ? If this is not true in general, then does assuming some further hypothesis like $R$ is an integral domain or say $R$ is local, would imply $R$ is Noetherian ?