I was trying to calculate the length of the $A_{\mathfrak{p}}$-module $A_{\mathfrak{p}}/\mathfrak{p}A_{\mathfrak{p}}$. It seems that the only $A_{\mathfrak{p}}$-submodule must be the zero module, but I can't figure out why. I have a feeling that the proof is really simple but I am not seeing it. I would appreciate any enlightening.
To give context, I was trying to calculate the order of vanishing of $z$ in $\mathbb{A}_{\mathbb{C}}^1$ at the origin.
You know that there is inclusion preserving one-one correspondence between the set of all prime ideals $I$ of $A$ with $I\cap(A-P)=\emptyset$ and the set of all prime ideals of $A_P$. Therefore $PA_P$ is the unique maximal of $A_P$. So $k=A_P/PA_P$ is a field. Since $(PA_P)k=0$, $A/PA_P$-submodules of $k$ and $A_P$-submodules of $k$ are same. So $l_{A_P}(k)=l_{k}(k)=1$.