Prove (projective$\Rightarrow$flat) using Lambek's criterion

Question

Lambek's criterion states that (left) $R$-module $M$ is flat if and only if $M^{*} = \mathrm{Hom}_{\mathbb{Z}}(M, \mathbb{Q}/\mathbb{Z})$ is an injective (right) $R$-module. Using this, I want to prove that every projective $R$-module is flat. To prove this, I have to show that for any projective $R$-module $M$, $M^{*}$ is injective, but I don't know how to do this. I tried to use exactness of functors but I failed.

Answer 1

If $M$ is projective then $M\oplus N=F$ is free for some module $N$. Then $M^*\oplus N^*=F^*$. If we can prove $F^*$ is injective, then it follows that $M^*$ is injective (direct summands of injectives are injective). As $F$ is a direct sum of copies of $R$, then $F^*$ is a direct product of copies of $R^*$. But $R^*$ is injective (Lambek) and so $F^*$ is injective (direct products of injectives are injective). Then $M^*$ is injective, and so $M$ is flat (Lambek).