If $K$ is a field, $A=K[x_1,...,x_n,...]$ (countably many variables) and $p=(x_1,...,x_n)\subset A$, I'm trying to prove or disprove that $A_{p}$ is noetherian.
I'm trying to argue that if $J\subsetneq A_p$, there exists $I\subset p$ such that $J=IA_p$ and the fact that $I\subset p$ allows us to conclude that $I$ is finitely generated.
Is this true? How can I see this?