Let be a commutative ring with 1.
- Prove that the tensor product of two free -modules is free.
- Prove that the tensor product of two projective -modules is projective.
Suppose and are free -modules; we can assume (since is commutative) that and for some nonempty index sets and . By this previous exercise, binary tensor products essentially commute with direct sums. Thus, . Thus is free.
Now suppose and are projective. Then there exist and so that and are free. By the previous argument, is free. Now is free, so that is projective.