Non freely presented module

Question

I would like an example of a commutative (unital) ring $R$ and an $R$-module $M$ which has no free presentation, i.e. there are no sets $I, J$ with an exact sequence $$\bigoplus_J R \to \bigoplus_I R \to M \to 0$$

Any module $M$ has a surjection from a free module, namely $\bigoplus_M R \to M, e_m \mapsto m$, but the kernel might not be free. Notice however that the map $\bigoplus_J R \to \bigoplus_I R$ is not required to be injective.

Answer 1

There's no such module. If the kernel of your map $\bigoplus_M R \to M, e_m \mapsto m$ is $ K $, then $ K $ itself is an $ R $-module, which means you can also find a surjection $ \bigoplus_K R \to K, e_k \mapsto k $. Then, the sequence

$$ \bigoplus_K R \to \bigoplus_M R \to M \to 0 $$

is exact.