Prove R/J is noetherian

Question

I have a noetherian ring $R$ and $J\triangleleft R$. Then $R/J$ is noetherian?

I suppose $I\triangleleft R/J$ and $\tilde{I}=\{r\in R:r+J \in I\}$.

$\tilde{I}\triangleleft R$, $R$ is noetherian so exist $a_1,....,a_n \in \tilde{I}: \tilde{I}=Ra_1+\cdots+Ra_n$

Can I say $I=\tilde{I}/J$?

Answer 1

All ascending chains of ideals in $R/J$ correspond to ascending chains of ideals in $R$ and so must stop.