Exercise 6, § 4

Let $k$ be a field, $B$ the polynomial ring $k[X, Y]$ and $A$ the submodule $k[X^2, XY, Y^2]$ of $B$. (a) Show that $A$ is an integrally closed Noetherian domain and that $B$ is a finitely generated $A$-module. (b) Show that the ideal $p = AX^2 + AXY$ of $A$ is a prime ideal of height 1 (and hence divisorial) but that the $B$-module $p \otimes_A B$ is a torsion module $\neq 0$ and hence not reflexive; the ideal $pB$ of $B$ is not divisorial, the canonical mapping $p \otimes_A B \to pB$ is not injective and $p$ is not a flat $A$-module.