I am studying the theorem that every Dedekind domain is integrally closed. Here's the setup.
Let $D$ be a Dedekind domain, $F$ its field of fractions and let $u$ be an element of $F$ that is $D$-integral (there exists a monic polynomial $f(x)=x^n+a_1x^{n-1}+...+a_n$ such that $f(u)=0$). Then, there exists a faithful $D[u]$ submodule of $F$ that is finitely generated as $D$ module. This submodule is $M=D1+Du+...+Du^{n-1}$.
I now wish to prove that this fractional ideal $M$ satisfies $M^2=M$, but I am having difficulties.
Would you help me, please? Thank you in advance.