Definition of subintegral extensions

Question

In his paper "On Seminormality", Swan defined subintegral extensions as follows:
An extension of commutative rings $A\subseteq B$ is subintegral if:
(1): $B$ is integral over $A$.
(2): $\mathrm{Spec}(B) \to \mathrm{Spec}(A)$ is bijective and $\kappa(\mathfrak{q} \cap A) \to \kappa(\mathfrak{q})$ is an isomorphism for all $\mathfrak{q}$ in $\mathrm{Spec}(B)$, where $\kappa$ denotes residue field.

My question is whether (2) implies (1).

Answer 1

No -- let $A$ be a ring, choose an element $f \in A$ and set $B := A/(f) \oplus A[f^{-1}]$. Then $A \to B$ satisfies (2) but does not satisfy (1) in general, for example if $A$ is normal and $f$ is (nonzero and) not a unit.