Give an example of a finitely generated $R$-module $M$ (for some commutative ring $R$) that is not projective and is not finitely presented.
I was able to find an example of a finitely generated $R$-module that is not projective; if $A$ is a nonzero finite abelian group, then $A$ is not projective over $\mathbb{Z}$. However, it seems that these are finitely presented.
Note: Here I say that a module is finitely presented if and only if there exists an exact sequence
$F_0 \rightarrow F_1 \rightarrow M \rightarrow 0$
where $F_0$ and $F_1$ are free with finite bases.
Best Answer
If $I\subset R$ is an ideal which is not finitely generated, then $R/I$ is not finitely presented as $R$-module, and not projective as well. (If $R/I$ is projective, then $I$ is a direct summand, so $I$ is generated by an idempotent.)