TITLE: Inverse Image of Maximal Ideals
QUESTION [19 upvotes]: Given a map of commutative rings with unit, it is often the case that the inverse image of a maximal ideal is not maximal.  For example, consider the inclusion $\mathbb{Z} \subseteq \mathbb{Q}$.
However, it is well-known that the inverse image of a maximal ideal under a map of finitely generated algebras over an algebraically closed field is maximal.
Are there other examples where we see this same behavior?  For example, 

Is the inverse image of a maximal ideal under a map of finitely generated $\mathbb{Z}$-algebras maximal?

REPLY [7 votes]: Yes, because $\mathbb{Z}$ is a Hilbert-Jacobson ring.  See e.g. $\S 12.2$ of these notes.