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
$$M f.g. \iff \mathbf{Hom}(M,-) \text{ preserves infinite sums?}$$



Hom functor and arbitrary coproducts
Yes, a projective module is small if and only if it is finitely generated.


(A module is called *small,* if its covariant Hom functor preserves coproducts.)


For a proof, see Proposition II.1.2 in H. Bass, *Algebraic K-Theory,* 1968. 


