Let $A,B$ be two Dedekind domains such $A\subset B$. On denote by $K$ and $L$ their quotient field respectively. One assumes que $[L:K]$ is finite. Let $I$ be an ideal of $A$. Suppose that $IB=a B$ with $a\in A$. Can one assert that $I=aA$?
Thanks in advance for any answer.
