Spec of an injective ring map contains minimal primes in its image?

Question

Let $f\colon A \rightarrow B$ be an injective ring homomorphism. One knows (from EGA I, 1.2.7 or elsewhere) that the image of $\mathrm{Spec}(f)$ is dense. Does that image necessarily contain all the minimal primes of $A$ (i.e., does it contain the generic points of the irreducible components of $\mathrm{Spec} A$)?

If this is true, then it is likely to be somewhere in Bourbaki or EGA but I cannot find it. I am unable to invent a counterexample though: the claim follows from the density if $B$ has finitely many minimal primes (e.g., is Noetherian) and I am having trouble thinking of other $B$.

Answer 1

This is one of the coolest tricks in commutative algebra.

Let $A\subset B$ be a subring. Take a minimal prime $p\subset A$. Then the localization $A_p$ has a unique prime ideal. It's enough to show that $B_p$ is non-zero, but localization is exact!