Let $R = k[x,y]$ , $I = (x,y)$ , $k$ is a field.

I want to prove that :

1) $x \otimes y - y \otimes x \neq 0 $ in $I \otimes\_{R} I$

2) $x \otimes y - y \otimes x $ is a torsion element

My thoughts: to prove that $x \otimes y - y \otimes x \neq 0 $ in $I \otimes\_{R} I$ probably I should find a bilinear map $$\phi : I \times I \to R$$ such that $\phi(x,y) \neq \phi (y,x)$ , but which one?