Determining whether a certain element in a tensor product is zero
Solution 1:
Hint: You can regard $k$ as a $k[x,y]$-module via the map $k[x,y]\to k$. Then, consider the map $I\times I\to k$ defined by
$$f(ax+by+\text{higher order},cx+dy+\text{higher order})=ad-bc$$
Check that $f$ really is a $k[x,y]$-bilinear map $I\times I\to k$. Note though that
$$f(x,y)=1\qquad f(y,x)=-1$$