Exercise 9, § 4

Let $k$ be a field and $A$ the polynomial ring $k[X, Y]$. On the $A$-module $M = A^2$ consider the linear form $f$ such that $f(e_1) = X, f(e_2) = Y((e_1, e_2))$ being the canonical basis). Show that the kernel $L$ off is a monogenous free $A$-module, but that the quotient $M/L$ (which is isomorphic to an ideal of $A$) is not reflexive.