Example of a flat module that is neither projective nor injective We knew that some counter-examples: $\mathbb{Q}$ is a flat $\mathbb{Z}$-module but not a projective $\mathbb{Z}$-module; $\mathbb{Z}$ is a flat $\mathbb{Z}$-module but not an injective $\mathbb{Z}$-module.
So, is there a flat module but neither projective nor injective?

Here is a simple example based on your given examples, in the category of $\mathbb{Z}$-modules: $\mathbb{Q}$ is flat but not projective, $\mathbb{Z}$ is flat but not injective, so put them together: $\mathbb{Q}\oplus\mathbb{Z}$ is flat, since the direct sum of flat modules is flat. It is not projective, since a direct summand of it is not projective, and it is not injective, since it is not divisible.
