There is something I don't understand in Neukirch's Algebraic number theorey. He said that:
"every ideal is a finitely generated $\mathbb Z$-module by (2.10) and therefore a fortiori a finitely generated $\mathcal O_K$-module."
I know that every ideal is a finitely generated $\mathbb Z$-module but I fail to understand why then it is a finitely generated $\mathcal O_K$-module. Could someone tell me something? Appreciate that.