I'm working through an exercise in Hilton/Stammbach's A Course in Homological Algebra that asks if the direct product of projective modules is always projective, and I'm running into trouble giving an elementrary proof that $M = \prod_{i=1}^\infty \mathbb{Z}$ is not projective. This thread gives a clever argument, but it still relies on the (nonobvious) fact that a submodule of a free $\mathbb{Z}$-module is free, which is proven later in the book than this problem is given.
Is an even more elementary proof possible, which does not use the fact that free and projective $\mathbb{Z}$-modules are the same, or that submodules of free $\mathbb{Z}$-modules are free? I'm thinking along the lines of demonstrating some surjective map $A \to M$ which does not have a right inverse, but I can't figure out a suitable module to pick for $A$.
