Hom functor and arbitrary coproducts

Question

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

Answer 1

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.