Principal non-prime ideal whose extension is prime

Question

Let $A \subset B$ be a finite extension of integral domains. Let $(a)$ be a principal ideal in $A$. If $(a)^e = aB$ is a prime ideal of $B$, does it follow that $(a)$ is prime in $A$ ?

I have a related question, but the given example doesn't work.

Answer 1

I don't think so. Take $A = k[x,xy,xy^2,y^3] \subset B = k[x,y]$. The ideal $xA$ is not prime, but $xB$ is; $x(xy^2) = (xy)^2$, but $xy \notin xA$ in $A$.