Let $K$ and $L$ be number fields with $K \subset L$, and let $R = \mathbb{A} \cap K$ and $S = \mathbb{A} \cap L$. Part (c) asks us to characterize the ideals of $S$ such that $I = (I \cap R) S$.
The same question is asked here: Algebraic number theory, Marcus, Chapter 3, Question 9. Although the problem is solved, I cannot understand the solution there.
The first two parts of the question (or the results I known) are
- Let $I, J$ be ideals in $R$. If $IS \mid JS$ then $I \mid J$.
- For each ideal $I$ in $R$, we have $I = IS \cap R$.
I am wondering if part (c) is related to these results.
I will show that $I$ satisfies the condition if and only if it is of the form $JS$, where $J$ is an ideal in $R$.
Suppose $I = JS$ for some ideal $J$ in $R$. Then by part (b) of the problem, $I\cap R = JS\cap R = J$. This implies, $(I\cap R)S = JS = I$.
Conversely, suppose $(I\cap R)S = I$. It suffices to show that $I\cap R$ is an ideal in R, but this is easy to see.