Criterion for proving flatness

Question

I am trying to show that:

$M$ is a flat $A$ module iff for all ideals $I$ of $A$, which are finitely generated, the $A$-linear map $I \otimes M\longrightarrow M$, taking $x \otimes y\longrightarrow xy$ is injective.

Please help.

Answer 1

Here's a reference: Proposition 3.58 of Rotman's 'An Introduction to Homological Algebra', second edition.

Answer 2

Let $M$ have this property. Then $I \otimes M \to M$ is injective for every ideal $I$ of $R$. To see this, use that tensor products are compatible with directed colimits and that directed colimits of monomorphisms are monomorphisms.

Now let $N \to N'$ be a monomorphism, we want to show that $N \otimes M \to N' \otimes M$ is a monomorphism. By a colimit argument, similar to the one above, we may assume that $N'$ is generated by finitely many elements over $N$. By induction it suffices to consider the case of a single generator, i.e. $N' = N + \langle n \rangle$. Now let $I=\{r \in R : rn \in N\}$. This is an ideal of $R$ with $N' = N \oplus_I R$ (pushout). It follows that $N' \otimes M = (N \otimes M) \oplus_{I \otimes M} M$. By the hypothesis $I \otimes M \to M$ is a monomorphism. It remains to remark that monomorphisms of $R$-modules are stable under pushouts.