Exercise 4, § 4

Let $k$ be a field and $A$ the polynomial ring $k[X, Y]$. (a) Let $(e_1, e_2)$ be the canonical basis of the $A$-module $A^2$ and let $E$ be the sub-$A$-module of $A^2$ generated by $(X - Y)e_1, e_1 + Xe_2$ and $e_1 + Ye_2$; let $F$ be the monogenous submodule of $E$ generated by $(X - Y)^2e_1$. Show that the $A$-module $M = E/F$ is not the direct sum of its torsion submodule and a torsion-free module. (b) Show that the torsion $A$-module $A/AXY$ is not a direct sum of monogenous submodules of the form $A/\mathfrak{p}_i^n$, where the $\mathfrak{p}_i$ are prime ideals of height 1.