Lambek theorem ($M\text{ flat}⟺M^{\ast }\text{ injective}$)

A module over a commutative ring is flat exactly when its character module is injective. For a submodule inclusion $N^{\prime }\hookrightarrow N$, the character adjunction identifies restriction ${\mathrm{Hom}}_A(N,M^{\ast })\to {\mathrm{Hom}}_A(N^{\prime },M^{\ast })$ with the dual of $N^{\prime }{\otimes }_{A}M\to N{\otimes }_{A}M$. The injective cogenerator of abelian groups detects whether the latter map is injective. Thus the two extension/exactness conditions are equivalent.