Isomorphism from $R[x,y] / (x, y - x^2)$ to $R$

Question

I know that by the third isomorphism theorem $R[x,y] / (x, y - x^2)$ is isomorphic to $R$. Therefore, $(x, y - x^2)$ should be the kernel of a homomorphism from $R[x,y]$ to $R$ by the first isomorphism theorem. Since $x$ belongs to the kernel, it must be that $x$ goes to $0$. However, I am struggling to see where $y$ should go.

Thanks!

Answer 1

Hint: Notice that $$(x,y-x^2)=(x,y-x^2,y-x^2+x^2)=(x,y)$$