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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?

2 Answers 2