The Stacks project

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

1. $A$ is local with maximal ideal $\mathfrak m$,

2. $A \to B$ is a finite¹ ring map,

3. $A \to B$ is injective (we think of $A$ as a subring of $B$).

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