Suppose $R$ and $S$ are domains, $S$ is integral over $R$, $R$ is integrally closed, $p_1$ and $p_2$ are primes in the domain $R$, $p_1$ contains $p_2$, $q_1$ is a prime in the domain $S$ and lying over $p_1$, we can prove that $p_2 S_{q_1}\cap R=p_1$,so $p_2S_{q_1}\cap S$ is lying over the prime $p_2$ as we wanted in the proof of the Going-down theorem. Is $p_2S_{q_1}\cap S$ also a prime? I find that many books just play a trick of "ec" to around it,but can we prove it directly?
A generated question: Suppose $R$ and $S$ are domains, $S$ is integral over $R$, $p$ is a prime in $R$, in which situation $pS$ is also a prime in $S$?(As we know, $(2)$ is prime in $\Bbb{Z}$, but not a prime in $\Bbb{Z}[i]$)