This is exercise 25 in Chapter 10, Section 5 of Dummit and Foote.

Essentially we are trying to prove that A is a flat R-module iff for every finitely generated ideal I of R, the map from A tensor I -> A tensor R = A induced by the inclusion I into R is again injective. (equivalently A tensor I = AI contained in A).

First we need to prove that if A is flat then A tensor I -> A tensor R is injective. This isn't difficult.

The next step is very tricky for me:

If A tensor I -> A tensor R is injective for every finitely generated ideal I, prove that A tensor I -> A tensor R is injective for every ideal I.

Show that if K is any submodule of a finitely generated free module F then

A tensor K -> A tensor F is injective.

Then I have to decduce that the same is true for any free module F.

Any ideas?