= Lambek theorem
{c}
{title2=$M\text{ flat}\iff M^*\text{ injective}$}
A <module> over a commutative <ring> is <flat> exactly when its <character module> is <injective>. For a submodule inclusion $N'\hookrightarrow N$, the character adjunction identifies restriction $\operatorname{Hom}_A(N,M^*)\to\operatorname{Hom}_A(N',M^*)$ with the dual of $N'\otimes_AM\to N\otimes_AM$. The <injective cogenerator of abelian groups> detects whether the latter map is injective. Thus the two extension/exactness conditions are equivalent.
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