Let be a ring and let be a unital right -module. Prove that is flat if and only if for all injective left -module homomorphisms where is finitely generated, the map is injective.
Certainly if is flat, then every such map is injective.
Suppose now that is a right -module such that for all injective left -module homomorphisms with finitely generated, is injective. Let be an arbitrary injective module homomorphism. Suppose . Now there exists a finitely generated submodule containing all of the . (For example, the submodule generated by the .) The restriction of to is injective by our hypothesis, and certainly . Thus . In particular, , so that is injective.
Thus is flat.