Exercise 111.58.3 (0D5I)—The Stacks project

Table of contents
Part 9: Miscellany
Chapter 111: Exercises
Section 111.58: Commutative Algebra, Final Exam, Fall 2016
Exercise 111.58.3 (cite)

Exercise 111.58.3. Let $A \to B$ be a ring map such that

$A$ is local with maximal ideal $\mathfrak m$,
$A \to B$ is a finite ¹ ring map,
$A \to B$ is injective (we think of $A$ as a subring of $B$).

Show that there is a prime ideal $\mathfrak q \subset B$ with $\mathfrak m = A \cap \mathfrak q$.

[1] Recall that this means $B$ is finite as an $A$-module.

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