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

From Commalg

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 is an integral extension of commutative unital rings. Then consider the induced map on spectra:

that sends a prime ideal of to its contraction in .

Then the following are true:

- If is a maximal ideal of , then is maximal in
- If is maximal in , then is maximal in

## 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.