Use of Grobner Basis to Compute Relations with respect to Quotient Polynomial Ring

Question

I think I might have phrased the question poorly, but I have a somewhat basic question about how to use Grobner bases in a particular example. Suppose I have some algebra $$R=\mathbb{C}[x_1,x_2^3,x_3^3,x_1x_2],$$ and the surjective map of $R$-modules $$ R\oplus R\rightarrow R\cdot x_2+R\cdot x_3^2,$$ where $e_1=(1,0)\mapsto x_2$ and $e_2=(0,1)\mapsto x_3$. Is there a way to systematically determine the kernel of the map? For example, I know that $(-x_3^3)\cdot x_2+(x_2x_3)\cdot x_3^2=0$, but is there a way to get all such relations?