Question 9 in Marcus book.
Let $K$ and $L$ be the number field such that $K\subset L$ and let $R,S$ be their algebraic integers, respectively.
a) Let $I$ and $J$ be ideals in $R$, and suppose $IS|JS$. Showw that $I|J$.
(Suggestion(from the book): Factor $I$ and $J$ in primes in $R$)
b) Show that for each ideal $I$ in $R$, we $I=IS\cap R$.
(Set $J=IS\cap R$ and use a)
c)characterise those ideals $I$ of $S$ such that $I=(I\cap R)S$.
Now, I need to understand something , if $P$ is any prime ideal in $R$, then what I can say about $PS$? I meant if $P$ is any prime ideal in the factorisation of $I$ , then is $PS\subset I$???
How to show $J=IS\cap R$?
Is there any website or book explain these concepts in more details and examples.
I really need to understand the concepts of this question.