Kernel of a substitution map

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.

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