# Checking if quotient ring is Noetherian or Artinian over a module

Given a quotient ring $k[t,w]/w^2$, and the lemma that if $A \subset B$ is a R-submodule then $B$ is Noetherian iff both $A$ and $B/A$ are Noetherian. Can we see if $k[t]$ is Noetherian over itself and see if $k[w]/w^2$ is Noetherian over $k[t]$ and would this tell us if $k[t,w]/w^2$ is Noetherian over a $k[t]$-module?

• Can you use somehow if R is Noetherian then R[x] is Noetherian and then use the fact that quotient of Noetherian is Noetherian. – Sunny Rathore Feb 17 '18 at 10:51
• Well I was thinking that $R = k[t, w]/w^2$ wasn't noetherian as a $R-$module, but since I need to know if R is noetherian as a $k[t]$- module and not an $R[t]$-modules I was thinking that I cannot use what you've suggested. Can you point out the flaw in my thinking please? – user901823 Feb 17 '18 at 20:09