Finding the kernel of a multiplication map Consider the ideal $I=(x,y) \subset R=\mathbb{C}[x,y]$ and $\mathbb{C}$ as the $R$-module $R/I$. I am asked to find the kernel of the multiplication map $I \otimes_R I \rightarrow I$ as a submodule of $I$. I know this is the ideal generated by $x\otimes y-y\otimes x$, but I don't know how to prove it. Any suggestions? 

Note that elements of $R$ can move across the tensor symbol, so every element in $I \otimes_R I$ can be written in the form $x \otimes f(x, y) - y \otimes g(x, y)$.  That this element is in the kernel means $xf = yg$ so $x \ | \ g$ and $y \ | \ f$.  Now we have $x \otimes yf' - y \otimes xg'$ and $xyf' = xyg'$.  Thus $f' = g'$ and our element is in the form $f'(x, y)(x \otimes y - y \otimes x)$.
