Let $R=k[v,x,y,z]/I, I=<v^2,z^2,xy,vx+xz,vy+yz,vx+y^2,vy-x^2>$,and let $f:R^2 \rightarrow R^2$ denote the map given by the matrix $M$, \begin{array}{cc} v & y \\ x & z \\ \end{array} I guess that there is a module $N$ such that $Cokerf \cong N\oplus k$, but I don't know how to prove it. Any comment is welcome. Thanks a lot!

Let $R=k[v,x,y,z]/I$, with $I=\langle v^2,z^2,xy,vx+xz,vy+yz,vx+y^2,vy-x^2\rangle$,and let $f:R^2 \rightarrow R^2$ denote the map given by the matrix $$M=\begin{pmatrix} v & y \\ x & z \end{pmatrix}$$ I guess that there is a module $N$ such that $\operatorname{coker}f \cong N\oplus k$, but I don't know how to prove it. Any comment is welcome. Thanks a lot!