= Solution
For an ideal $I$, the canonical multiplication map
$$
\mu_I:I\otimes_AM\longrightarrow M,\qquad i\otimes m\longmapsto im
$$
has image $IM$. Consequently the asserted isomorphism is the natural surjection $I\otimes_AM\twoheadrightarrow IM$ being injective. This is the <flatness criterion using ideals>, whose proof follows from the extension criterion above.
Under the character <Tensor-hom adjunction>, the dual map $\mu_I^*$ identifies with
$$
\operatorname{Hom}_A(A,M^*)\longrightarrow\operatorname{Hom}_A(I,M^*).
$$
Indeed, for $\phi\in M^*$, its restriction corresponds to $i\otimes m\mapsto\phi(im)$. If $M^*$ is <injective>, this restriction is surjective, so the cogenerator criterion makes $\mu_I$ injective. Conversely, if every $\mu_I$ is injective, every restriction map in this display is surjective. The proved <Baer criterion> then makes $M^*$ <injective>. Thus
$$
\boxed{(2)\Longleftrightarrow(3).}
$$
In particular, (1) also implies (3) directly by tensoring $I\hookrightarrow A$. No finite-generation condition on $I$, and no Noetherian condition on $A$, has been inserted.
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