Example of a compact module which is not finitely generated

Question

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?

Answer 1

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.