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.

  • 1
    $\begingroup$ 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

1 Answer 1


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 $S$.

  • $\begingroup$ (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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.