Exercise 2, § 1

Let $K$ be a commutative field and $A$ the sub-K-algebra of the polynomial algebra $K[X, Y]$ generated by the monomials $Y^k X^{k+1} \ (k \geqslant 0)$. Show that $XY$ is such that $A[XY]$ is contained in a finitely generated $A$-module but $XY$ is not integral over $A$.