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Let $M$ be a finitely generated $R$-module. It's easy to check that, in this case, $\mathbf{Hom}(M,-)$ preserves infinite sums. Now suppose that $M$ is projective. Is the reciprocal true? That is, if $M$ is a projective module, is it true that $$M f.g. \iff \mathbf{Hom}(M,-) \text{ preserves infinite sums?}$$

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Yes. It is exactly as you stated it. In general you have that the covariant hom functor $\text{Hom}(M,-)$ preserves arbitrary sums, if $M$ is finitely generated and you get an equivalence when the module is projective. Projectiveness is really necessary as shown for example in

R. Rentschler: Sur les modules M tels que Hom(M,-) commute avec les sommes directes. C. R. Acad. Sci. Paris 268 (1969), 930–932.

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