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 $\mathfrak m = A \cap \mathfrak q$.

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