I am working through the problems in Dummit and Foote and can't seem to work out (or even begin to start) 15.4.15(c) (p.727):
Let $R = \mathbb{Z}[\sqrt{-5}]$ be the ring of integers in the quadratic field $\mathbb{Q}(\sqrt{-5})$ and let $I$ be the prime ideal $(2, 1 + \sqrt{-5})$ of $R$ generated by $2$ and $1 + \sqrt{-5}$. Recall that every nonzero prime ideal $P$ of $R$ contains a prime $p \in \mathbb{Z}$.
(c) Prove that $I_P \cong R_P$ as $R_P$-modules for every prime ideal $P$ of $R$ but that $I$ and $R$ are not isomorphic $R$-modules.
It appears that we can begin by showing that if $P$ is a prime ideal of $R$ not containing 2, then $I_P = R_P$
And furthermore, if $P$ is a prime ideal of $R$ containing 2 then $P=I$ and $I_P = (1+ \sqrt{-5})R_P$. This is as the authors and Trevor suggest. However, I am unsure of how to do this.
When $P \ne I$ you should be able to show that $I_P = R_P$. When $P = I$ then $I_I$ is the maximal ideal of the DVR $R_I$. In a DVR the maximal ideal is principal so $I_I = \pi R_I$ for some element $\pi$. As an $R_I$-module, this is isomorphic to $R_I$ via the map $x \mapsto \pi x$.