Solution
= Solution
The <prime ideal correspondence for localization> identifies primes of $B_{\mathfrak p}$ with primes $\mathfrak q$ of $B$ disjoint from $A\setminus\mathfrak p$. Passing to the quotient by $\mathfrak pB_{\mathfrak p}$ retains exactly those containing $\mathfrak pB_{\mathfrak p}$. Thus the image is the <fiber of the map on spectra>:
$$
\left\{\mathfrak q\in\operatorname{Spec}B:\mathfrak q\cap A=\mathfrak p\right\}.
$$