Following Eisenbud's definition, I say that a prime of a ring $A$ is associated to a nonzerodivisor if it is associated to a principal ideal generated by a nonzerodivisor.
I'm trying to prove that if $A\subseteq B$ is a finite extension of noetherian domains s.t. $A$ is normal, then $B$ is normal, with the additional hypothesis that for each prime $\mathfrak{q}\subset B$, if $\mathfrak{p}=\mathfrak{q}\cap A$, then $\mathfrak{p}B_\mathfrak{q}=\mathfrak{q}B_\mathfrak{q}$.
I'm trying to use Serre's criterion which in the case of an integral domain $A$ reduces normality to the condition that $(\ast)$ every prime associated to a nonzerodivisor is s.t. $\mathfrak{p}A_\mathfrak{p}$ is principal.
If I manage to prove that given such an ideal $\mathfrak{q}\subset B$, its contraction $\mathfrak{p}$ is also associated to a nonzerodivisor (or it has height 1), then since $\mathfrak{p}A_\mathfrak{p}$ is principal, also $\mathfrak{q}B_\mathfrak{q}$ is principal and I would be done.
I know that in general contraction of principal ideals is not principal, but may be with the additional hypothesis of $A\subseteq B$ being finite and injective this is true (?) and it would be more than enough to conclude my proof; the only thing I managed to prove is that if $(b)\subset B$ is principal, then $(b)\cap A\ne (0)$ since $b$ satisfies an integral equation of the type $b^n+a_{n-1}b^{n-1}+\cdots+a_0=0$ (and I can suppose $a_0\neq 0$ by choosing $n$ minimal) and one has $0\neq a_0\in(b)\cap A$.
Can someone give me a hint about how to procede?