(All rings assumed to be commutative and unital)

Given a ring $R$, a prime ideal $\mathfrak{p}$ of $R$, and an extension ring $S$ (the algebra map $R\to S$ is injective), are there any nontrivial sufficient conditions for $rad(\mathfrak{p}S)$ to be prime? How about when $S$ is integral over $R$? How about when it is finitely presented as an $R$-module?

EDIT: Qing Liu's comment means that I should be even more specific.

Assume the third question above ($S$ is finitely presented as an $R$-module.).

Suppose further that there exists an element $f \in R$ such that $R_f$=$S_f$. Then even further, suppose $f \in \mathfrak{p}$.

EDIT 2: It is now open season for adding any extra conditions on $S$ that you want. Please don't add stupid conditions like S=R. Note that the condition about $f$ is actually a condition on $S$.