Let $A$ be a finitely generated $K$-algebra over a field $K$. A typical problem is to find a maximal ideal $\frak{m}$ such that $f\notin\mathfrak{m}$ and it does not coincide (or contains) another ideal $I$.
The solution would be to use Krull's theorem that every ring with a multiplicative identity has a maximal ideal and consider the non-empty localization of quotient $$ A'=\left(A/I\right)_{f} $$ This ring $A'$ has a maximal ideal, which corresponds to a maximal ideal in $A$, which satisfies the required conditions.
From what I've seen, this "trick" is usually applied to finitely generated $K$-algebras over a field $K$. I would like to know if there is any reason that limits its application, i.e. can't it be applied to infinitely generated K-algebras?
Maximal ideals in $B_f$ don't have to pull back to maximal ideals of $B$ when $B$ is not of finite type. (It may even happen that $B_f$ is a field and $B$ is not.)