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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.

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    $\begingroup$ Because an $\mathcal O_K$-module is also a $\mathbf Z$-module (by restriction of scalars), and ‘he who can do the more can do the less’. $\endgroup$
    – Bernard
    Oct 13, 2018 at 13:12

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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 $S$.

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  • $\begingroup$ (not needed for your particular question, but this generalizes to the case that $f: R \to S$ is some not necessarily injective map and we view $M$ as an $R$-module via $f$.) $\endgroup$
    – hunter
    Oct 13, 2018 at 13:11
  • $\begingroup$ yeah……seems that I asked a dumb question, thanks anyway. $\endgroup$
    – DL Qian
    Oct 13, 2018 at 13:17

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