$(x) \supseteq (x^2) \supseteq (x^4) ... $ is a descending chain in $A[x]$ that is not stationary if $A[x]$ is not Artinian.
What is an example of a module such that $Ann(x) \subseteq Ann(x^2) \subseteq Ann(x^3)...$ is stationary if $A[x]$ is Noetherian?
I dont see the inclusion from the definition of $Annhilator(x)= (0:x)$. I tried looking at $R=A[x]/x^4$. Then $(x^3) \in Ann(x)$, and $(x^2) \in Ann(x^2)$