I am trying to prove (or disprove) the following:
For a non-Noetherian ring $R$ and a non-finitely generated ideal $I$ the $R$-module $R/I$ is not finitely presented.
I don't think that it is enough to consider s.e.s. $0\rightarrow I\rightarrow R\rightarrow R/I \rightarrow 0$ and argue that $I$ is not finitely generated so we are done. When I consider a general surjection $0\rightarrow \ker(f)\rightarrow R^n\rightarrow R/I\rightarrow 0$ I don't know how to proceed. I could show that $\ker f$ contains $I$ as a submodule, that's all.
If this is not true in general, I wonder if it holds for the case where $R=k[x_1,...]$ with infinite generators and $I=(x_1,...)$ the maximal ideal.