If $A,B$ are commutative rings with $1$, $p$ is a prime ideal in $A$ and $f:A\rightarrow B$ makes $B$ an $A$-algebra, I want to know if it is possible to define the localization $B_p$ of $B$ at the extension of $p$.
I suspect it is possible, because if we view $B$ as an $A$-module, we obtain the $A_p$-module $B_p$. This is isomorphic to $B\otimes_A A_p$, which carries the structure of an $A$-algebra. Thus, $B_p$ is an $A$-algebra.
However, it is not clear to me that $B\setminus p$ is a multiplicative subset of $B$, so I'm stuck.