Let be a ring with 1 and let be a family of free unital left -modules. Prove that is free.
Let be a free basis for each . We can assume that the are disjoint. (For example, isomorphically replace each by .) Let . Let denote the free -module on .
We have the natural inclusion . We also have a natural inclusion , where , where if and otherwise. By the universal property of free modules, there is a unique -module homomorphism such that . We claim that is an -module isomorphism.
Suppose . Say . Now . Since is free on , we have for all , for all . Thus , so that , and thus is injective.
Now suppose . Since only finitely many of the are nonzero, . Certainly , so that is surjective.
Thus , and thus is free as an -module.