Let $R$ be a ring and $M$ be an $R$-module.

Definition: $M$ is called compact if $\text{Hom}_R(M,-)$ commutes with direct sums, that is, if for any set $I$ and any $I$-indexed family of $R$-modules $\{N_i\}_{i\in I}$ the canonical map of abelian groups $$\bigoplus\limits_{i\in I}\text{Hom}_R\left(M,N_i\right)\longrightarrow\text{Hom}_R\left(M,\bigoplus\limits_{i\in I} N_i\right)$$ is an isomorphism.

Example: Any finitely generated $R$-module is compact.

Proposition: If $R$ is Noetherian, any compact module is finitely generated.

I'm looking for an example witnessing that this is not true without $R$ being Noetherian:

Question: What is an example of a ring $R$ and a compact, but not finitely generated $R$-module?

  • $\begingroup$ This question seems to say that regarless of $R$ being Noetherian, compact $\iff$ finitely presentend $\implies$ finitely generated. But the two definitions of "compact" are different, are they equivalent? $\endgroup$ – Najib Idrissi Feb 14 '15 at 20:46
  • $\begingroup$ @NajibIdrissi: Yes, I suspect the compactness property I'd like to understand is strictly weaker than the one in the link you posted - but thanks for that, it's good to have a reference for the case of preservation of arbitrary directed limits. $\endgroup$ – Hanno Feb 14 '15 at 20:51
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    $\begingroup$ Yes, in the linked MO question they are called "sumpact" (sum compacts), and it is stated that it's a weaker property than "compact" ($\hom(M,-)$ commutes with filtered limits). In fact this answer by Jeremy Rickard answers your question, I believe. $\endgroup$ – Najib Idrissi Feb 14 '15 at 20:59

This MO answer by Jeremy Rickard answers your question:

A fairly simple explicit example of a "sumpact"* module that is not f.g. is as follows.

Let $R$ be the ring of functions from an uncountable set $X$ to, say, a field $k$. Let $M$ be the ideal of functions with countable support.

Then it's very easy to show that $M$ isn't f.g., and fairly easy to show that it is "sumpact", using no set theory beyond the fact that a countable union of countable sets is countable.

* A sumpact module is a $R$-module $M$ such that $\hom_R(M, -)$ commutes with arbitrary direct sums – what is called "compact" in the question.

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    $\begingroup$ This is a CW answer so that the question doesn't remain unanswered, but if someone is willing to post a different answer that's fine by me. I have to admit I don't understand what the downvote is for? $\endgroup$ – Najib Idrissi Feb 14 '15 at 21:05
  • $\begingroup$ I don't understand either @ downvote... Thank you very much for your help! (+1 of course) $\endgroup$ – Hanno Feb 14 '15 at 21:12
  • $\begingroup$ @Najib Idrissi Could you describe explicitly the example please? I asked the related question math.stackexchange.com/questions/1187256/… $\endgroup$ – Loronegro Mar 27 '15 at 20:11

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