I'm reading Neukirch's book about Algebraic number theory and I totally fail to understand something. At one point he proves that the ring of integers $\mathcal{O}_K$ of a number field $K$ is noetherian in the following manner:
"every ideal is a finitely generated $\mathbb{Z}$-module by (2.10) and therefore a fortiori a finitely generated $\mathcal{O}_K$-module."
Now 2.10 is the claim that for an integrally closed, principle ideal domain A, F its field of fractions, L a finite separable extension of F and B the integral closure of A in L, we have that "every finitely generated B-submodule $M\ne 0$ of L is a free A-module of rank $[L:F]$".
I fail to see how 2.10 is used here. I guess in our case $F=\mathbb{Q}, L=K, A=\mathbb{Z}, B=\mathcal{O}_K$. So I know that every finitely generated $\mathcal{O}_K$ submodule (in other words, ideal) is a free $\mathbb{Z}$-module, which is additional information; but how can I assume that the ideal is finitely generated in the first place? That's what I need to prove!