Let be a principal ideal domain. Prove that a right unital -module is flat if and only if it is torsion free.
Suppose is flat as a right -module. Given nonzero, define by . Because is a commutative ring, is a left module homomorphism. Because is a domain, is injective. Because is flat, is injective. Now suppose , with . In , we have , so that . Since is injective, we have . Recall that via the multiplication map . Thus we have , and so is torsion free as a right -module.
Conversely, suppose is torsion free as a right -module, and let be a nonzero ideal. Since is a PID, say with nonzero. Note that the map given by is a module homomorphism. Moreover, has a trivial kernel since is torsion free- so is injective. Now consider the map given by ; this is a module homomorphism. Certainly is surjective, and moreover since is a domain, is injective. Thus is a module isomorphism. Now the homomorphism is also an isomorphism since and . Recall that and are isomorphic as abelian groups via the mappings and given by and . Consider the map .
Now . Thus we have , and so . Thus is injective.
By the flatness criterion for modules (proved here), is flat as a right -module.