Extensions and contractions of prime ideals under integral extensions
For example if $R\subseteq S$ is flat: the extension then actually is faithfully flat and thus for every ideal $I\subseteq R$ we have $IS\cap R=I$. Applying this to $I=P^n$ yields $$P^n = SP^n\cap R = (SP)^n \cap R = Q^n \cap R.$$
However flatness is not a frequent property of integral extensions ...