Let $\varphi:\Bbb C[x,y]\to\Bbb C[t]$ be homomorphism that sends $x\to t$ and $y\to t^2$. This is surjective map with kernel $(y-x^2)$. The correspondence theorem relates ideals $I$ of $\Bbb C[x,y]$ that contain $y-x^2$ to ideals $J$ of $\Bbb C[t]$ by $J=\varphi(I)$ and $I=\varphi^{-1}(J)$. Since $J$ will be principal ideal $(p(x))$, each ideal $I$ of $\Bbb C[x,y]$ containing $(y-x^2)$ will be $(y-x^2,p(x))$.
Now let's consider $\Phi:\Bbb R[x,y]\to\Bbb R[t]$ be the homomorphism that is the identity on real numbers and sends $x\to t^2, y\to t^3$. We see that the kernel is $(y^2-x^3)$. Can we have some similar info abut kernels of $R[x,y]$ containing $(y^2-x^3)$? (In the aforementioned example, we had the luxury of $x\to t$.)