Prove that no nontrivial, divisible -module is projective. Deduce that is not projective.
We begin with a lemma.
Lemma: Let be a free -module. Then . Proof: Suppose is a basis for indexed by a set , and let . For all , there exist such that and all but finitely many are zero. Since is free on , if , we have for all so that . Thus . Hence .
Now suppose the nontrivial, divisible -module is projective. Then there exists a module such that is free. By the lemma, . Since is injective and is a principal ideal domain, for all . So we have . Note, however, that this module contains , and since is nontrivial, we have a contradiction. So is not projective.
In particular, since is nontrivial and divisible (as we showed here), it is not projective as a -module.