Every element has power in subring implies bijective on spectra

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 $A$ is a subring of a commutative unital ring $B$, with the following property: for any $b\in B$, there exists an integer $n$ (possibly dependent on $b$) such that $b^{n}\in A$).

Then, the map on spectra:

$Spec(B)\to Spec(A)$

is a bijection.