Exercise 3, § 2

Let A be the ring $K[X, Y]$ of polynomials in two indeterminates over a field K. (a) Consider in A the principal ideals $b = (X)$, $c = (Y)$, which are free A-modules and whose intersection $b \cap c = (XY)$ is also free. Show that $a = b + c$ is not a flat A-module, although a is torsion-free (cf. Algebra, Chapter III, § 2, Exercise 4). (b) In the A-module $A^2$, let R be the submodule consisting of the elements $(x, -x)$ where $x \in a$. In the A-module $A^2/R$, let M, N be the submodules which are the images of the factor submodules of $A^2$; show that M and N are isomorphic to A, but that $M \cap N$ is not a flat A-module.