Checking if quotient ring is Noetherian or Artinian over a module

Question

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?