I have a problem with my lesson about generated module: Find a family $\{M_\alpha\}_\Delta$ of R-modules each of which is finitely generated but $\oplus_\Delta M_\alpha$ is not finitely generated.


closed as unclear what you're asking by Leucippus, John Wayland Bales, Namaste, steven gregory, Shailesh Sep 13 '17 at 0:11

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You mean like $\oplus _{i=1}^\infty F$ for a field $F$, as an $F$-module?

Each $M_\alpha=F$ is a cyclic $F$ module, and the sum of infinitely many is not finitely generated.

  • $\begingroup$ it is just a direct product of R-modules $M_\alpha$ $\endgroup$ – Hoàng Sep 12 '17 at 14:47
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    $\begingroup$ @Hoàng I don't understand your comment. There isn't any direct product mentioned in your question or mine... This example appears to suit your requirements. $\endgroup$ – rschwieb Sep 12 '17 at 15:04
  • $\begingroup$ sorry because of my bad so i dont understand your comment before edited. so is there an example with $\Delta$ is finite? $\endgroup$ – Hoàng Sep 12 '17 at 15:11
  • $\begingroup$ @Hoàng No, in that case, the sum is obviously generated by the union of the generators of the finitely many $M_\alpha$. $\endgroup$ – rschwieb Sep 12 '17 at 15:21
  • $\begingroup$ okay i understand it, thank you for answer ^^ $\endgroup$ – Hoàng Sep 12 '17 at 15:25

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