A finitely generated flat module that is not projective

Let’s see an example of a finitely-generated flat module that is not projective!

What does this provide a counterexample to?

If $R$ is a ring that is either right Noetherian or a local ring (that is, has a unique maximal right ideal or equivalently, a unique maximal left ideal), then every finitely-generated flat right $R$-module is projective.

So what happens if we drop the Noetherian and local hypotheses?

The Example

Let $R = \prod_{j=1}^\infty F_j$ be an infinite product of fields and let $I = \oplus_{i=1}^\infty F_j$ be the ideal that is the direct sum of all the fields. Then the module $R/I$ is finitely generated. It is also flat, because $R$ is von Neumann regular and in such rings, every module is flat. Why is it not projective?

To see that it is not projective, consider the exact sequence
$$0\to I\to R\to R/I\to 0.$$ If $R/I$ were projective, that would mean that the map $R\to R/I$ splits, which gives a direct sum decomposition $I\oplus R/I\xrightarrow{\sim} R$ where the composition of the map $I\to I\oplus R/I\to R$ is the inclusion $I\to R$. The image of $R/I$ then corresponds to a nonzero ideal in $R$. But any nonzero ideal intersects $I$, so such a splitting is impossible.

This example is part of my new counterexamples project.

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