Currently studying for qualifying exams and came across the following problem:
Give an explicit example of a ring $R$ (commutative with identity) and a surjection $\psi: M \rightarrow N$ of finitely generated $R$-modules such that $\ker(\psi)$ is not a finitely generated $R$-module.
The given hint is to let $R = \mathbb{Z}[x_1, x_2, ... ]$, however I am confused how such a construction would yield finitely generated submodules? It seems clear that we want to invoke the general strategy of picking an ideal $I$ of $R$ and defining the natural surjection $$\phi : R \rightarrow R/I$$ If we let $I = \langle x_1, x_2, ...., \rangle$ then $\ker(\phi) \cong I$ and $I$ is not finitely generated, however $R$ is also not finitely generated. Thus, we do not satisfy the problem statement? Any help on this with the given hint would be greatly appreciated!