We knew that the direct sum of a family of projective modules is a projective module, and the direct product of a family of injective modules is also injective.
My question is, is the infinite direct product of an infinite family of projective modules also projective? If it is, please give an elegant proof, or else please give some counter-examples.
Thank for reading my question.
Best Answer
Note: The question originally asked about infinite direct sums.
If you meant "infinite direct product", on the other hand, then the answer is "no." Consider $\mathbb{Z}$-modules; then a module is projective if and only if it is free. So $\mathbb{Z}$ itself is projective, but $$\prod_{i=1}^{\infty}\mathbb{Z}$$ is not free, hence not a projective $\mathbb{Z}$-module.
Here's the answer to the original question:
The direct sum of free modules is free, even if the family is infinite.
Assume that $\{P_i\}_{i\in I}$ is a family of projective modules; for each $i$, there exists $M_i$ such that $P_i\oplus M_i$ is free. Then $$\left(\bigoplus_{i\in I}P_i\right) \oplus \left(\bigoplus_{i\in I}M_i\right) \cong \bigoplus_{i\in I}(P_i\oplus M_i)$$ is a direct sum of free modules, hence free. Since there is a module $M$ (namely, $M=\oplus M_i$) such that $(\oplus P_i)\oplus M$ is free, it follows that $\oplus P_i$ is projective.
That is, any direct sum of projective modules is projective.