Given an $R$-module $M$, we define its character module as $M^{\ast}=Hom_{\mathbb{Z}}(M, \mathbb{Q/Z})$. I have already proved that $M$ is a flat module iff $M^{\ast}$ is an injective module. Now I am stucked in proving the following two conclusions. If anyone can help me out, thanks a lot.
(1) The natural map $M \rightarrow M^{\ast \ast}$ is injective, which may not be an isomorphism.
(2) Every flat $R$-module is a filtered colimit of free $R$-module.