Exercise 2, § 3

Let $K$ be a field and $B$ the quotient of the polynomial ring $K[X, Y]$ in two indeterminates by the ideal $a$ of $K[X, Y]$ generated by $X^2$ and $XY$; denote by $x$ and $y$ the images in $B$ of the elements $X, Y$; let $A$ be the subring $K[x]$ of $B$. Show that the elements $x$ and $y^n$ ($n \geq 0$) form a basis of $B$ over $K$ and that $B$ is not integral over $A$. On the other hand, there exists in $B$ no element which is algebraically free over $A$. Deduce a counter-example to Corollary 1 to Theorem 1 of no. 1 when $A$ is not an integral domain.