Let be a ring with 1 and let be a family of (right, unital) -modules. Prove that is flat if and only if each is flat.
We begin with a lemma.
Lemma: Let be a family of (unital) -module homomorphisms. Then given by is injective if and only if each is injective. Proof: Suppose is injective. If , and letting denote the inclusion or , then , so that . Thus each is injective. Conversely, suppose each is injective; then if , we have for each , so that for each . So , and thus is injective.
Suppose each is flat. Now let and be left unital -modules and let be a module homomorphism. Since each is flat, is injective. Using the lemma, is injective. In this previous exercise, we constructed group isomorphisms and . We claim that . To that end, note that , as desired. Thus is injective, and so is flat.
Conversely, suppose is flat. Let and be (unital, left) -modules and a module homomorphism. Note that is injective. Recall the homomorphisms and from the previous paragraph; is an injective module homomorphism mapping to , and in fact . By the lemma, each is injective, so that each is flat.