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.
Best Answer
If $R \subset S$ are rings and $M$ is an $S$-module which is finitely generated as an $R$-module, then it is finitely generated as an $S$-module too. Indeed, if $m_1, \ldots, m_n$ is a list of generators, then everything in $M$ is a finite $R$-linear combination $\sum r_i m_i$, so it's also a finite $S$-linear combination because the coefficients $r_i$ are also in $S$.