TITLE: Kernel of a substitution map
QUESTION [6 upvotes]: Suppose $R=k[x,y,z]$ and $S=k[t]$. Consider the map $f:R\to S$ s.t. $f(x)=t$, $f(y)=t^2$ and $f(z)=t^3$. I suspect the kernel of this map is the ideal $(y-x^2,z-x^3)R$. 

It's clearly contained in the kernel, but I am not sure how to prove the reverse inclusion.

REPLY [2 votes]: Since $\mathfrak I = (y-x^2,z-x^3)$ is contained in the kernel, your map corresponds to a map $R/\mathfrak I\to S$. Show that this map is injective.