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

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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.