I'm trying to solve this question but I'm stucked:
If a ring $R$ satisfies the following two conditions:
i) For every maximal ideal $M$ of $R$, if $S = R\setminus M$ then $S^{-1}R$ is Noetherian,
ii) For every $x \in R$, $x\neq 0$, the set of maximal ideals that contains $x$ is finite,
then $R$ is Noetherian.
I tried to show that every ideal of $R$ is finitely generated but I don't get anything with this.
Thank you for any help.
