As the title suggests, I would like to decompose the ideal $I:=(x^2-y, z-1)\vartriangleleft k[x,y,z]$ into primary ideals.
I suspect it may be prime already, if $I= \ker G$, where $$G: k[x,y,z] \to k[s], \\ f(x,y,z) \mapsto f(s,s^2,1).$$
At least we must have $I \subseteq \ker G$, since $x^2 -y \mapsto s^2 -s^2 =0$ and $z-1 \mapsto 1-1 =0$. Does the converse inclusion hold?
If not: how would I go about decomposing this ideal?