Every element has power in subring implies bijective 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.