Equivalence of Finiteness of Class number to Property of Ideals of Algebraic Integers I was reminded of this problem by a recent question of Regenbogen. It is not too difficult to prove that since every number field has finite class group, that every ideal in the ring of algebraic integers is either principal or infinitely generated. I've been unable to prove that these statements are actually equivalent, and this is what I'd be interested to know.
In particular, if they are equivalent, then this may provide a different way of proving finiteness of class number.
 A: In the statement of the question you meant "every ideal is finitely generated". 
In any event, these two types of finiteness (of ideals as modules and of the ideal class group) are not at all equivalent.  For example, you can replace integral closures of $\mathbf Z$ in finite extensions of the rationals with integral closures of $F[x]$ in finite extensions of $F(x)$, where $F$ is a field, and thus speak about ideal class groups 
in this other setting.  There the ideal class groups can be infinite, even though the ideals are still finitely generated.  To take a concrete example, consider the integral closure of $\mathbf C[x]$ in $\mathbf C(x,y)$ where $y^2 = x^3 - x$.  That integral closure is a Dedekind domain and its class group is infinite; in fact the ideal class group is isomorphic to the group of complex points on the elliptic curve $y^2 = x^3 - x$, which as a group looks like $\mathbf C/\mathbf Z[i]$.
Finiteness of class groups is somewhat special (not unique to it, but still special) to the setting of rings of algebraic integers and not valid in general Dedekind domains. 
In a general Dedekind domain, any ideal has at most 2 generators and this is unrelated to the size of the class group.
