I'm asked to prove that the ideal $I=(x,y)$ in $R=k[x,y]$ is not a flat R-module.
My approach was to use the exact sequence $$0\rightarrow I \to R \to R/I \to 0$$ to induce a non injective map $$I\otimes I\to R\otimes I$$ because the element $x\otimes y-y\otimes x$ is sent to $0$ by the inclusion.
But I need to show that such element is not zero in the domain. My intuitive approach is that such element is nonzero because in $I\otimes I$ I'm not allowed to move $x$ or $y$ "inside" the tensor because $1\notin I$. But I cannot formalise it properly. So how to prove that such tensor is not zero?