Module $M_{ 5 }$

Description:

Let $a,b$ be generators for the module satisfying the following relations (with $X,Y$ the images of $x,y$ in $R=F_2[x,y]/(x,y)^2$): $aY=bX=0$ and $aX=bY$. This produces a module $M$ which is a 3-dimensional $F_2$ space spanned by $\{a,b,aX\}$.