Let $K$ be a field. I am trying to fit the $R=K[x,y]$-modules, $M=(K[x,y] \times K[x,y])/\left<(x,0),(y^2,x),(0,y^3)\right>$, and $N=K[x,y]/\left<x^2,xy,y^3\right>$ into a non-split exact sequence $0 \rightarrow M \rightarrow L \rightarrow N \rightarrow 0$, for some $R$-module $L$, but I am having trouble doing that.
Any element in $N$ is of the form $a_0+a_1x+a_2y+a_3y^2$, whereas any element in $M$ is of the form $(F(y),G(x))$, and we have two possibilities, i.e., $ G(x)= k \in K$, in which case $F(y)$ is a polynomial in $y$ of any order, and $F(y)=a_1y+a_2$, in which case $G(x)$ is a polynomial in $x$ of any order.
Any hints would be appreciated.