# Integral extension implies inverse image of max-spectrum is max-spectrum

This article gives the statement, and possibly proof, of a fact about how a property of a homomorphism of commutative unital rings, forces a property for the induced map on spectra
View other facts about induced maps on spectra

## Statement

Suppose $f:R \to S$ is an integral extension of commutative unital rings. Then consider the induced map on spectra:

$f^*: Spec(S) \to Spec(R)$

that sends a prime ideal $P$ of $S$ to its contraction $P^c = f^{-1}(P)$ in $R$.

Then the following are true:

• If $\mathfrak{m}$ is a maximal ideal of $S$, then $f^*(\mathfrak{m})$ is maximal in $R$
• If $f^*(P)$ is maximal in $R$, then $P$ is maximal in $S$

## Proof

The key idea here is the following fact: for a finite extension of integral domains, one of them is a field if and only if the other is.