= Flatness criterion using ideals
{title2=$I\otimes_AM\xrightarrow{\sim}IM\quad\text{for every ideal }I$}
The arrow is the canonical multiplication map, not an arbitrary abstract module isomorphism. Its inclusion into $M$ is $\mu_I:I\otimes_AM\to M$. Dualizing identifies $\mu_I^*$ with the restriction $\operatorname{Hom}_A(A,M^*)\to\operatorname{Hom}_A(I,M^*)$. All these maps are surjective precisely when $M^*$ is injective, by the <Baer criterion>. The <injective cogenerator of abelian groups> detects injectivity of $\mu_I$, and the <Lambek theorem> gives flatness. No Noetherian or finite-generation hypothesis is needed.
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