I am trying to translate a section of Wolfgang Krull's report "Idealtheorie". At one point (Section $7$ on Quotient Rings) I believe that he makes something like the following statement:
Suppose for example that $K$ and $L$ are finite algebraic number fields with $K$ contained in $L$ and let $R$ resp. $S$ be the ring of all integers in $K$ resp. $L$. Then the set of all contracted ideals in $R$ consists of all ideals of $R$.
My question is "Have I got this right (my German is not great). i.e. is this true of rings of integers?" Clearly it is not true in general.
I don't know if your translation is correct, but your statement about ideals is correct: the ring extension $R\subseteq S$ is flat, because for every prime ideal $q$ of $S$ the localization $S_q$ is a flat $R_p$-module, $p:=q\cap A$, since $R_p$ is a discrete valuation domain and $S_q$ is torsion-free. Moreover the natural map $\mathrm{Spec}(S)\rightarrow\mathrm{Spec}(R)$ is surjective, hence $R\subseteq S$ is faithfully flat.
A faithfully flat ring extensions $R\subseteq S$ has the property $IS\cap R=I$ for every ideal $I$ of $R$.