I am doing Exercise 4.30 in Eisenbud's Commutative Algebra with a View Toward Algebraic Geometry. I will copy the problem statement here:
Exercise 4.30: Suppose that $k$ is a Noetherian ring such that
$*)$ for every finitely generated $k$-algebra $R$ and maximal ideal $P \subset R$ the $k$-algebra $R/P$ is finite over $k$.
Show that for every reduced finitely generated $k$-algebra $R$ and prime ideal $Q \subset R$ we have $Q = \bigcap P$, where the intersection runs over all primes $P$ of $R$ such that $R/P$ is finite over $k$ [error: should be "all primes $P$ of $R$ containing $Q$ such that $R/P$ is finite over $k$"]. (Hint: If $f \in R$, $f \notin Q$, we must find a prime $P$ such that $R/P$ is finite over $k$ and $f \notin P$. Consider a maximal ideal in the $k$-algebra $R[f^{-1}]$ and its intersection with $R$.)Deduce in particular that Theorem 1.6 [the Nullstellensatz] (for a given field $k$) follows if we prove that $*)$ holds for $k$.
(Square brackets are my annotations; additionally, all rings are commutative and have multiplicative identity.)
Using Zorn's lemma, I found a maximal ideal $\mathfrak{m}$ of $R[f^{-1}]$ containing the image of $Q$ in $R[f^{-1}]$. The preimage of $\mathfrak{m}$ under the natural map from $R$ to $R[f^{-1}]$ is a prime ideal. I don't think it is maximal (I remember that maps $A \to B$ of rings induce maps from the spectrum of $B$ to the spectrum of $A$, but not necessarily maps from the maximal spectrum of $B$ to the maximal spectrum of $A$). This prevents us from directly using condition $*)$. Also, $R[f^{-1}]$ is not necessarily finitely generated, and we also have the issue of the map $R \to R[f^{-1}]$ not being injective.
How do I connect the maximality of $\mathfrak{m}$ in $R[f^{-1}]$ with the finitely-generated property of $R$ in order to use property $*)$?
(I know there is a related question at Exercise 4.30 of Eisenbud, but I don't understand the second-to-last sentence of the answer.)
