I've come across the structure Theorem for fin. gen. Modules over a Dedekind domain several times now. It was formulated to us the following way:
Let $R$ be a Dedekind domain. For every element $\alpha \in C(R)$, let a representative $I_{\alpha}$ in the group of fractional ideals be chosen. Then, to a fin. gen. $R$-Mod $M$ there are unique natural numbers $r$ and $s$, $\alpha \in C(R)$ with $\alpha = 0$ if $s = 0$, and proper nonzero ideals $I_r \subset … \subset I_1$ such that
$M \cong R/I_1 \oplus… \oplus R/I_r, \text{if }s = 0$
$M \cong R/I_1 \oplus … \oplus R/I_r \oplus R^{s-1} \oplus I_{\alpha}$, if $s > 0$
Now I want to give a description of the finitely generated module over the Dedekind domain according to the structure theorem. In each case the elements of the class group are listed for you, each given by means of a representative ideal.
Dedekind domain $\mathbb{Z}[\sqrt{79}]$, class group of order 3, representatives $(1), (3,\sqrt{79}+1), (3, \sqrt{79}-1)$.
The Module $M = I \oplus I$, where $I = (3, \sqrt{79}+1)$.
I actually don't know where to begin. I also have not found any examples on the web. Any of those would be very appreciated.
Best Answer
To do your exercise, you need one extra piece of information, a theorem of Steinitz:
Let $r$ and $s$ be non-negative integers, and let $I_1,\ldots,I_r$ and $J_1,\ldots,J_s$ be ideals in your Dedekind domain $R$. Then one has an isomorphism $\bigoplus_{m=1}^r I_m\cong \bigoplus_{n=1}^sJ_n$ of $R$-modules if and only if $r=s$ and the equality $\prod_m [I_m] = \prod_n [J_n]$ holds in the class group of $R$.
For a proof, see Curtis and Reiner, Representation Theory of Finite Groups and Associative Algebras, Wiley (1962), Section 22.
You should be able to solve your exercise with this extra information.