In this case, $A_\mathfrak{p}$ is $S^{-1}A$, where $S = A \setminus \mathfrak{p}$. This came up in a question about showing that $\kappa(\mathfrak{p}) \cong A_p/\mathfrak{p}A_\mathfrak{p}$. I managed to solve it (mostly) using an earlier proof that I did about localization commuting with quotients.
However, I don't feel like I'm done because I'm not understanding the notation $\mathfrak{p}A_\mathfrak{p}$. What does it mean? I think it's a specific case of some more general notation about ideals, but I've forgotten what it refers to. Is it perhaps the image of $\mathfrak{p}$ in the localization map from $A \to S^{-1}A$?
Thanks for the help!
It's the extended ideal of $\mathfrak p$ through that map, i.e. the ideal generated by the image of $\mathfrak p$ through the map $A\to A_{\mathfrak p}$. Or, the ideal of all the fractions in $A_{\mathfrak p}$ the numerator of which is in $\mathfrak p$.