Exercise 31, § 7

Let $K_0$ be a commutative field, $A$ the polynomial ring $K_0[X, Y]$ in two indeterminates over $K_0$, which is an integral domain, and $E$ the ideal $AX + AY$ of $A$. Show that in the tensor product $E \otimes_A E$ the element $X \otimes Y - Y \otimes X$ is $\neq 0$ and that $XY(X \otimes Y - Y \otimes X) = 0$ (consider the $A$-bilinear mappings of $E \times E$ into the quotient module $A/E$).*